A quasi-linear theory is presented for how randomly forced, barotropic velocity fluctuations cause an exponentially-growing, large-scale (mean) magnetic dynamo in the presence of a uniform shear flow, $vec{U} = S x vec{e}_y$. It is a kinematic theory for the growth of the mean magnetic energy from a small initial seed, neglecting the saturation effects of the Lorentz force. The quasi-linear approximation is most broadly justifiable by its correspondence with computational solutions of nonlinear magneto-hydrodynamics, and it is rigorously derived in the limit of large resistivity, $eta rightarrow infty$. Dynamo action occurs even without mean helicity in the forcing or flow, but random helicity variance is then essential. In a sufficiently large domain and with small wavenumber $k_z$ in the direction perpendicular to the mean shearing plane, a positive exponential growth rate $gamma$ can occur for arbitrary values of $eta$, the viscosity $ u$, and the random-forcing correlation time $t_f$ and phase angle $theta_f$ in the shearing plane. The value of $gamma$ is independent of the domain size. The shear dynamo is fast, with finite $gamma > 0$ in the limit of $eta rightarrow 0$. Averaged over the random forcing ensemble, the mean magnetic field grows more slowly, if at all, compared to the r.m.s. field (or magnetic energy). In the limit of small Reynolds numbers ($eta, u rightarrow infty$), the dynamo behavior is related to the well-known alpha--omega {it ansatz} when the forcing is steady ($t_f rightarrow infty$) and to the incoherent alpha--omega {it ansatz} when the forcing is purely fluctuating.
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