  We prove the existence of a compact random attractor for the stochastic
Benjamin-Bona-Mahony Equation defined on an unbounded domain. This random
attractor is invariant and attracts every pulled-back tempered random set under
the forward flow. The asymptotic compactness of the random dynamical system is
established by a tail-estimates method, which shows that the solutions are
uniformly asymptotically small when space and time variables approach infinity.
