We prove uniqueness for extended real-valued lower semicontinuous viscosity solutions of the Bellman equation for L(infinity)-control problems. This result is then used to prove uniqueness for Isc solutions of Hamilton-Jacobi equations of the form -u(t) + H(t, x, u, -Du) = 0, where H(t, x, r, p) is convex in p. The remaining assumptions on H in the variables r and p extend the currently known results.