The bipartite Ramsey number $b(m,n)$ is the smallest positive integer $r$ such that every (red, green) coloring of the edges of $K_{r,r}$ contains either a red $K_{m,m}$ or a green $K_{n,n}$. We obtain asymptotic bounds for $b(m,n)$ for $m \geq 2$ fixed and $n \rightarrow \infty$.