The Roelcke uniformity on a topological group G is the greatest lower bound of the left and the right uniformities. Let H be an infinite-dimensional Hilbert space, and let U(H) be the unitary group endowed with the topology of pointwise convergence. We show that the completion of U(H) with respect to the Roelcke uniformity can be identified with the semigroup of all linear operators of norm ≤ 1 on H. This yields a new proof of Stoyanov’s theorem which asserts that the group U(H) is totally minimal.