Conventionally, though with justifications, space is said to begin at the Kármán line which is
100km from Earth surface, i.e., still pretty close. The atmospheric
pressure at this altitude drops to about 0.032 Pa (wikipedia), which is still a
lot more than outer space (less than $10^{-4}$ Pa according to wikipedia)
The phase diagram of water shows that, at this pressure level, water
can exist only as a solid or as vapor, depending on temperature, but
not as a liquid. The phase transition between solid and gaz at that
low pressure takes place near 200°K (around -73°C), which is not that cold.
So, if you drop in space a blob of water at room temperature and
pressure it will instanly start to evaporate (boil) and decompress.
Here I am not sure about what happens. There are accounts from
astronauts on the web that explain that the water (actually urine)
will first vaporise then desublimate into tiny crystals. But no
explanation of the actual physical phenomena that drive it.
My own reconstruction of what could happen (before I saw these sites)
is the following.
First the loss of pressure propagates very fast in the liquid (speed of
sound?) while loss of temperature (heat) propagates slowly (as all beer lovers
know from their fridge). So the boiling will essentially take place
uniformly in the whole liquid. Phase transition from liquid to gas
absorbs heat, and that is what will cool the water very quickly, as
it evaporates.
My guess is also that the energy loss will cool the water down to
sublimation temperature (solid-gas transition) before it all
evaporates, so that some parts of the liquid may be cooled down to
freezing before they have time to evaporate. But as boiling takes
place everywhere, it actually breaks the remaining water into tiny
fragments that cristallize, and possibly also collect some of the
vapor to grow.
Anyway, you apparently get snow.
But the cooling is due to evaporation, which is very fast,
much more than to radiation which has hardly any time to take place.
Numerical evaluation
We analyze what becomes of available heat to understand whether some water freezes directly. This is a very rough approximation as the figures used are
actually somewhat variable with temperature, but I cannot find the
actual values for the extreme temperature and pressures being
considered.
The specific latent heat of evaporation of water is 2270 kJ/kg. The
specific heat of water is 4.2 kJ/kgK Hence, evaporating 1 gram of
water can cool 2270/4.2 = 540 grams of water by 1°K, or 5.4 grams by
100°K which is about the difference between room temperature and water
(de)sublimation temperature in space. So my hypothesis that there is
not enough heat available to vaporize all the water is correct, as
only about one sixth of the water can be vaporized with the available
heat.
Out of 5.4g of water, 1g will evaporate, though may cool down to just
above the sublimation temperature of 200°K, while the remaining 4.4g
will be cooled to sublimation temperature without vaporizing,
yet. The remaining 4.4g cannot remain liquid, hence, one part freezes,
thus freeing some latent het for the other part to vaporize. The ratio
between the two part is inversely proportional to the specific latent
heat for freezing and vaporizing.
Latent heat for freezing is 334 kJ/kg.
The sum of both latent heat is 2270+334=2604 kJ/kg. These figure are
very approximate. As a sanity check, the latent heat of sublimation of
water is approximately 2850kJ/kg (wikipedia), which show that the
figures are probably correct within a 10% approximation.
The ratio divides the remaining 4.4g into approximately 3.8g that
freezes and 0.6g that evaporates, making it a total of 1.6g of
vaporized water.
So, skipping a quick calculation, we find that about 70% of the water freezes into some kind of snow, while the remaining 30% are vaporized. And it all happens rather quickly.
I was actually uneasy about this account of astronauts stories of
water boiling and then desublimating at once, because that would leave
us with all the heat to get rid of very quickly. How? Does anyone have a better
account?
A last remark is that there always will be some part of the water that
gets frozen. I thought initially that very hot water might provide enough heat to vaporize itself completely un low pressure. The critical point of liquid water is at 650°K (with a much higher pressure than you care to create in space: 22MPa), which is
only 450° above the sublimation temperature. But the water should be
cooled by 540° to provide enough heat to evaporate completely. So
the water temperature will drop to the sublimation threshold before
enough heat can be supplied to evaporate it completely. This problably
a very simplistic analysis, though. I leave the rest to specialists.
Best Answer
Water boils when the pressure is less than its vapour pressure (there is a table of vapour pressure vs temperature here).
At 20ºC the vapour pressure is 2339Pa, so if your balloon exerts a pressure greater than this the water won't boil. If the pressure exerted by the ballon is less than this, the water will start to boil and the steam generated will inflate the balloon. This will increase the pressure until the pressure has risen to 2339Pa, and at this point the water will stop boiling and you're left with water and steam in equilibrium.
If you heat water to 100ºC the vapour pressure rises to 1 atm (101325Pa), which is of course why water boils at 100ºC at sea level. However if you have a really strong balloon capable of exerting a pressure of greater than 1 atm the water won't boil at 100ºC even in a vacuum, and you'd need to raise the temperature above 100ºC to make it boil.