The Elliptic Logarithm Method has been applied with great success to the problem of computing all integer solutions of equations of degree 3 and 1 defining elliptic curves. We extend this method to include any equation f(u, v) = 0, where f € Z[u,v] is irreducible over Q. defines a curve of genus 1. but is otherwise of arbitrary shape and degree. We give a detailed description of the general features of our approach, and conclude with two rather unusual excimples corresponding to equations of degree 5 and degree 9.