The dynamics of an infinite one-dimensional system of hard rods is generalized to include the effects of a random background. Each rod follows a trajectory which is described by a generalized random function and when two rods collide they interchange velocities (and trajectories). An exact solution is obtained for the distribution ${f}_{s}(x,v,\frac{t}{{v}^{\ensuremath{'}}})$ which is the probability of finding a particle at $x$ with velocity $v$ at time $t$ that was at $x=0$ with velocity ${v}^{\ensuremath{'}}$ at $t=0$. The most interesting result is that in the long-time limit $p(x,t)$, which is the probability of finding a particle at $x$ at time $t$ that was at $x=0$ at $t=0$, is of the form ${t}^{\ensuremath{-}\frac{1}{4}}\mathrm{exp}(\ensuremath{-}\frac{{x}^{2}}{{t}^{\frac{1}{2}}})$. Thus, the spatial distribution does not become Gaussian and Fick's law is not valid. It is suggested that this qualitative behavior might be expected whenever single-file effects become important and that it is not dependent on the details of the one-dimensional hard-rod collisions which have been used in the derivation.