  Let G be a simple balanced bipartite graph on $2n$ vertices, $\delta =
\delta(G)/n$, and $\rho={\delta + \sqrt{2 \delta -1} \over 2}$. If $\delta >
1/2$ then it has a $\lfloor \rho n \rfloor$-regular spanning subgraph. The
statement is nearly tight.
