A numerical simulation algorithm that is exact for any time step \ensuremath{\Delta}tg0 is derived for the Ornstein-Uhlenbeck process X(t) and its time integral Y(t). The algorithm allows one to make efficient, unapproximated simulations of, for instance, the velocity and position components of a particle undergoing Brownian motion, and the electric current and transported charge in a simple R-L circuit, provided appropriate values are assigned to the Ornstein-Uhlenbeck relaxation time \ensuremath{\tau} and diffusion constant c. A simple Taylor expansion in \ensuremath{\Delta}t of the exact simulation formulas shows how the first-order simulation formulas, which are implicit in the Langevin equation for X(t) and the defining equation for Y(t), are modified in second order. The exact simulation algorithm is used here to illustrate the zero-\ensuremath{\tau} limit theorem. \textcopyright{} 1996 The American Physical Society.