The degree sequence of an n-vertex graph is d₀, ..., d_n-1, where each d_i is the number of vertices of degree i in the graph. A random graph with degree sequence d₀, ..., d_n-1 is a randomly selected member of the set of graphs on {0, ..., n-1} with that degree sequence, all choices being equally likely. Let \lambda₀, \lambda₁, ... be a sequence of nonnegative reals summing to 1. A class of finite graphs has degree sequences approximated by \lambda₀, \lambda₁, ... if, for every i and n, the members of the class of size n have lambda_in + o(n) vertices of degree i. Our main result is a convergence law for random graphs with degree sequences approximated by some sequence \lambda₀, \lambda₁, .... With certain conditions on the sequence \lambda₀, \lambda₁, ..., the probability of any first-order sentence on random graphs of size n converges to a limit as n grows.