Weight distributions of convolutional codes are important because they permit computation of bounds on the error performance. In this paper, we present a novel approach to computing the complete weight distribution function (WDF) of a convolutional code. We compute the weight distribution series using the generalized Viterbi Algorithm (GVA) and then find the minimum linear recursion relation in this series using the shift register synthesis algorithm (SRSA). The WDF follows from the minimum recursion. In order to generalize the use of the SRSA over certain commutative rings, we prove the key result that the set of finite recursions forms a principal ideal.