We report on measurements of the collisional properties of a mixture of $^{133}$Cs and $^{87}$Rb atoms in a magnetic trap at $mumathrm{K}$ temperatures. By selectively evaporating the Rb atoms using a radio-frequency field, we achieved sympathetic cooling of Cs down to a few $mumathrm{K}$. The inter-species collisional cross-section was determined through rethermalization measurements, leading to an estimate of $a_s=595 a_0$ for the s-wave scattering length for Rb in the $|F=2, m_F=2>$ and Cs in the $|F=4, m_F=4>$ magnetic states. We briefly speculate on the prospects for reaching Bose-Einstein condensation of Cs inside a magnetic trap through sympathetic cooling.