Let H _d (n,p) signify a random d-uniform hypergraph with n vertices in which each of the possible edges is present with probability p = p(n) independently, and let H _d (n,m) denote a uniformly distributed d-uniform hypergraph with n vertices and m edges. We establish a local limit theorem for the number of vertices and edges in the largest component of H _d (n,p) in the regime , thereby determining the joint distribution of these parameters precisely. As an application, we derive an asymptotic formula for the probability that H _d (n,m) is connected, thus obtaining a formula for the asymptotic number of connected hypergraphs with a given number of vertices and edges. While most prior work on this subject relies on techniques from enumerative combinatorics, we present a new, purely probabilistic approach.