Families of vortex equilibria, with constant vorticity, in steady flow past a flat plate are computed numerically. An equilibrium configuration, which can be thought of as a desingularized point vortex, involves a single symmetric vortex patch located wholly on one side of the plate. Given that the outermost edge of the vortex is unit distance from the plate, the equilibria depend on three parameters: the length of the plate, circulation about the plate, and the distance of the innermost edge of the vortex from the plate. Families in which there is zero circulation about the plate and for which the Kutta condition at the plate ends is satisfied are both considered. Properties such as vortex area, lift and free-stream speed are computed. Time-dependent numerical simulations are used to investigate the stability of the computed steady solutions.