What you are asking doesn't really fall into the framework of the SVM. There is some work on incorporating prior knowledge into SVMs (see e.g. here but these approaches are generally not on an example by example basis.
I can think of one way in which you could approach this, if you have a lot of samples. You could use the weights as probabilities for inclusion in random subsets. You would then learn the SVM on each subset, and your final classifier is then a linear combination of these subsets. This is a variation on bootstrapping, which normally works over subsets of the features (see e.g. here, and might be quite interesting to analyse.
[Edit 1]:
Based on the answers from Jeff and Dikran it occured to me that you can just incorporate into the SVM objective. Normally the primal form looks like:
$\min_{\mathbf{w},\mathbf{\xi}, b } \left\{\frac{1}{2} \|\mathbf{w}\|^2 + C \sum_{i=1}^n \xi_i \right\}$
subject to (for any $i=1,\dots n$)
$y_i(\mathbf{w}\cdot\mathbf{x_i} - b) \ge 1 - \xi_i, ~~~~\xi_i \ge 0 .$
but you could just include another vector of confidence values, e.g. $0 < \delta_i \leq 1, ~~~~i=1,\dots n$:
$\min_{\mathbf{w},\mathbf{\xi}, b } \left\{\frac{1}{2} \|\mathbf{w}\|^2 + \frac{C}{\delta_i} \sum_{i=1}^n \xi_i \right\}$
subject to (for any $i=1,\dots n$)
$y_i(\mathbf{w}\cdot\mathbf{x_i} - b) \ge 1 - \xi_i, ~~~~\xi_i \ge 0 .$
which would mean that instances with low probability would receive a greater penalty in the objective. Note that now the $C$ parameter performs two roles - as a regulariser and as a scaling factor for the confidence scores. This may cause its own problems, so it might be better to split it into two parts, but then of course you would have an extra hyperparameter to tune.
[Edit 2]:
This can be done with libSVM (MATLAB and Python interfaces are included). There is also code available in several languages for the SMO algorithm which can solve the SVM problem efficiently. Alternatively you could use an optimisation package, such as quadprog in matlab or CVX, to write a custom solver.
Best Answer
The term that you want to search for is "semi-supervised learning." Here's the wikipedia link, for convenience.
In a nutshell, you should think of this as "clustering with hints." You start with the labeled points, and get a sense of what they look like -- then you can make a good guess that other points that look like them also have that label, and finally, that points that don't look like them are labeled differently.
As always, there are lots and lots of tricks and theories and variations that go with this -- but given the breadth of your question, I think this will get you started in the right direction. Come back when your first results come through!