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In 1967, E. M. Stein proved that the Hilbert transform is bounded from $L^{\infty }$ to BMO. In 1976, Muckenhoupt and Wheeden gave an analogue of Stein’s result. They gave a necessary and sufficient condition for the boundedness of the Hilbert transform from $L_w^{\infty }$. We improve the results of Muckenhoupt and Wheeden’s and give a necessary and sufficient condition for the boundedness of the Hilbert transform from $BM{O}_w$ to $BM{O}_w$.