I think you're right: M(0) = 1 for any random variable X that you care to think of, because (by definition):
M(t) = E[e^(tX)] => M(0) = E[e^0] = E(1) = 1.
How did you find your expression for the m.g.f. - did you integrate by parts? If so, you might have started by writing something like:
M(t) = (1/2) * Integral(-1,1){e^(tx).(1-x) dx}
Notice that it doesn't make sense to integrate the "e^(tx)" bit up to (1/t)e^(tx) when t = 0 (because you end up dividing by zero). You may have to consider the case t = 0 separately (like above).