Do the derivatives of a moment generating function evaluated at 0 uniquely determine the distribution of a random variable?

Question

I understand that the moment generating function of a random variable uniquely determines that random variable. And that for an rv X, M(0)=1, M'(0)=E[X], M''(0)=E[X^2],... and so on.

My question is that if M'(0)=E[X] gives the general form of the expectation of say, a gamma random variable, then does that imply that X follows a gamma distribution?