We present a numerical solution of the Euler equations of gas dynamics for a weak-shock Mach reflection in a half-space. In our numerical solutions, the incident, reflected, and Mach shocks meet at a triple point, and there is a supersonic patch behind the triple point, as proposed by Guderley. A theoretical analysis supports the existence of an expansion fan at the triple point, in addition to the three shocks. This solution is in complete agreement with the numerical solution of the unsteady transonic small-disturbance equations obtained by Hunter & Brio (2000), which provides an asymptotic description of a weak-shock Mach reflection. The supersonic patch is extremely small, and this work is the first time it has been resolved in a numerical solution of the Euler equations. The numerical solution uses six levels of grid refinement around the triple point. A delicate combination of numerical techniques is required to minimize both the effects of numerical diffusion and the generation of numerical oscillations at grid interfaces and shocks.