  The conjecture of Wolmer Vasconcelos on the vanishing of the first Hilbert
coefficient $e_1(Q)$ is solved affirmatively, where $Q$ is a parameter ideal in
a Noetherian local ring. Basic properties of the rings for which $e_1(Q)$
vanishes are derived. The invariance of $e_1(Q)$ for parameter ideals $Q$ and
its relationship to Buchsbaum rings are studied.
