In order to find the mass of an electron experimentally, we find the value of e/m and we find the value of e. From these two values, we find the value of m.
Has any experiment been done, which measures the mass of an electron without finding its charge?
[Physics] Can mass of an electron be measured without measuring its charge
chargeelectronsmass
Related Solutions
The mass-to-charge ratio $m/e$ of the electron was first measured by J.J. Thomson, the discoverer of the electron, using cathode rays in 1897:
It should not be surprising that one may measure this ratio even without isolating "individual electrons" because the electric force acting on a charge may be written as $$ F = ma = Ee, \quad a = E\cdot \frac em $$ So what was left was just to measure the mass or charge separately. Millikan and Fletcher did the relevant oil drop experiment in 1909.
The electric force $F=Ee$ acting on a single drop with charge $e$, a single extra (or deficit) electron, may be calculated when it is set equal to the drag force from hydrodynamics, $6\pi e\eta v_1$. The viscosity $\eta$ is the most difficult thing to know but otherwise all quantities are known so $e$ may be calculated.
If one knows the charge and the ratio, one may calculate the mass as $m = e/ (e/m)$.
I'm not sure whether these theoretical ideas are is included in what you have in mind. They are only good (and the first , as far as I know, only in theory) for fundamental particles and not for measuring masses of everyday things, but here goes. The second - inference from cross coupling co-efficient between otherwise dispersionless, massless states - is actually the method we use to show that neutrinos have mass, but so far we haven't refined it enough to accurately measure that mass. Still, an inference that the rest mass is nonzero is still highly significant and counts for something IMO. Moreover, we may refine this method to give numbers in the future.
Method 1: Fundamental Particle Dispersion Relationships
This method is to infer the mass of a fundamental particle from experimentally measured dispersion relationships.
A possible fourth quality to add to your list is that mass measures what I call a fundamental particle's "stay-puttability". This is actually the generalisation $E^2 = p^2 c^2 + m_0^2 c^4$ the mass-energy equivalence you cite in disguise. (the equation is simply the pseudo-norm of the momentum 4-vector rewritten).
To look at this idea further, let's think of the Klein-Gordon equation for a lone, first quantised particle, which each spinor component of something fulfilling the Dirac equation must fulfill:
$$\left(-\hbar^2 \partial_t^2 + \hbar^2\,c^2 \nabla^2 - m_0^2\,c^4\right)\psi = 0\tag{1}$$
Hopefully you can pick out $E^2 - p^2 c^2 - m_0^2 c^4=0$ from the unwonted way I've written the equation: recall $i\hbar\partial_t$ is simply the LHS of the general Schödinger equation, so that, by the Schödinger equation, $\hat{H}$ and thus equivalent to the energy observable; also $-i\hbar\nabla$ is the momentum observable. Maxwell's equations can also be thought of as a kind of massless Dirac equation, so that the components of the potential four-vector also fulfill (1) and we can think of the photon as being included in this discussion.
For pure energy eigenstates, $i\hbar\partial_t = \hbar \omega$ and if we Fourier transform (1) into momentum space, we get from (1) the dispersion relationship for the fundamental particle:
$$\omega^2 = k^2\,c^2 +\frac{m_0^2\,c^4}{\hbar^2}\tag{2}$$
so that the group velocity is:
$$v_g = \frac{\mathrm{d}\,\omega}{\mathrm{d}\,k} = \frac{c}{\sqrt{1+\frac{m_0^2\,c^2}{\hbar^2\,k^2}}}\tag{3}$$
Massless particles must always be observed to be travelling at speed $c$, as shown by (3). They are always dispersionless. However, if $m_0$ is nonzero in (3) you can slow a particle down, or "make it stay put" by making the momentum $\hbar\,k$ very small. You can see now from (3) what I mean by mass measures a particle's "stay puttability".
So now you can in theory measure $k$ from matter diffraction experiments, or select for a narrow $k$ from a stream of particles whose mass you are trying to measure using a Bragg grating (for electrons or neutrons, read near-perfect matter crystal). Then you can presumably measure their velocities, within the bounds of the Heisenberg uncertainty principle, by using a matter version of something like a Fizeau-Foucault apparatus: i.e. a sequence of chopper wheels with angular displacements between their slits, so that only particles of a certain velocity, proportional to the chopper wheel angular speed, can make it through the chopper wheels. Then you vary the chopper speed to observe which speeds you detect particles at, and this will let you work out $v_g$. Knowing $v_g$ and $k$ now lets you work out $m_0$ from (3).
Method 2: Cross Coupling Co-efficient Measurement
This, as far as I can understand, is actually the method we use to know that neutrinos have mass. So far it is not very accurate: we can only infer nonzero mass but we haven't refined the method enough to say what that mass is. However, we may do so in the future. The beginning point of this discussion is the Dirac equation for the electron written in a particular way: we write the equations for the so-called Weyl spinors, which are a kind of circular polarisation for the electron:
$$\begin{array}{lcl}\partial\!\!\!/ \psi_L &=& -m\,\psi_R\\\partial\!\!\!/ \psi_R &=& +m\,\psi_L\end{array}\tag{4}$$
Maxwell's equations written in the same form are:
$$\begin{array}{lcl}\partial\!\!\!/ \psi_L &=& 0\\\partial\!\!\!/ \psi_R &=& 0\end{array}\tag{5}$$
That is, on comparing (4) and (5), the electron can be thought of as otherwise two massless, dispersionless particles, mutually tethered together by the cross term $m$; note the two first order equations are uncoupled in the Maxwell equation case. The first massless particle "tries" to zip off at the speed of light. Before this particle gets very far, the cross coupling term $m$ in (4) means that it changes into the other particle, which then also "tries" to zip off at lightspeed, only to be converted back to the first particle and the cycle repeats. This is the phenomenon that Schrödinger called the "Zitterbewegung" (German for quivvering motion) (can you say this word aloud without smiling? - I can't! It's a wonderful example of onomatopoeia). The nett result is that the mutually tethered system - the electron - has a rest mass: confined massless particles always have an inertia, as I discuss in my answer here.
Likewise for the neutrino. It used to be thought that the Weyl equation for the neutrino was the same as (4): three uncoupled, massless Weyl equations for the neutrino flavours. But we experimentally observe that a neutrino shifts between flavours as it propagates. Thus we know that there is a nonzero coupling co-efficient between the flavours, and therefore a mass. So flavour oscillation may in the future be another method for measuring mass.
Best Answer
Yes, the mass of the electron can be measured by the energy of the gamma rays involved in electron-positron annihilation. The mass of an electron is equal to the energy (divided by the speed of light squared) of one of those gamma rays, as measured in the center of momentum frame.
The measurement of gamma ray energies is known as Gamma-ray spectroscopy. The general idea is that the gamma ray is absorbed by some material, and the resulting ionization is detected in some way (cheaper detectors tend to use a scintillating material, while more expensive detectors use semiconductor technologies). The detector response is calibrated with a source of known gamma energies, and then the energy of another gamma ray can be measured.
The detector has a finite resolution, but the measurement can be enhanced by looking at extra features at the spectrum. For instance, there's a "Compton edge" that appears from gamma rays that Compton scatter rather than being absorbed, and an extra peak at twice the electron mass (when both gamma rays from the annihilation are absorbed).
That said, the relative uncertainty on the electron mass is only around $10^{-8}$, which is much better than you can get with standard gamma ray spectroscopy.
Some physics programs (Rutger's, for instance) have their students actually run this experiment in a lab course.