Numerically stable fast convolution algorithm?
There is the number theoretic transform where all calculations are done using integers which should completely get round any concerns about numerical stability, at the possible expense of some speed (see e.g. https://ccrma.stanford.edu/~jos/st/Number_Theoretic_Transform.html).  However this will only work for you if your numbers are rational and can be converted to integer representation easily.

There is also a whole range of asymptotically slower algorithms known as Toom-Cook multiplication http://en.wikipedia.org/wiki/Toom-Cook_multiplication. However, I am not sure it is true that multiplication via FFT calculations is necessarily numerically unstable. Do you have a reference for this claim?