The motion of an overdamped particle in a bistable potential U(x), driven quasimonochromatic noise (high-frequency, narrow-band noise), has been investigated by means of analog simulation. The escape rate from one potential well to another was found to be exponentially small compared to the reciprocal mean first-passage time to the top of the potential barrier. The logarithm of the quasistationary probability distribution was observed to fall extremely sharply at a particular value of x, quite close to the equilibrium position. Theory describing the nonanalytic dependence of this logarithm on the bandwidth of the noise is presented and shown to be in good agreement with experiment. Data are also presented for a symmetric monostable potential. In a certain parameter range, the quasistationary distribution is demonstrated to be independent of the form of such a potential.