Based on controlling the polarizations of the pump and Stokes beams in an optical fiber, a simple scheme for enhancing the stability of stimulated-Brillouin-scattering (SBS) slow light is proposed. By fusing a polarization-maintaining circulator (its fast axis is blocked) to each terminal of a polarization-maintaining fiber (PMF), we construct a slow-light element which ensures the two beams always having identical polarizations (along the slow axis of the PMF). Meanwhile, an experimental setup is established to study the slow-light performance. A tunable time delay of 0.87 ns/dB is demonstrated for a Gaussian pulse with a pulse width of 38 ns. The gain factor of the Brillouin gain in this element enhances three times as compared with that in the single-mode fiber. The experiment indicates that such a polarization-managed scheme significantly improves the stability of SBS interaction and thus that of the slow-light time delay.