An initial–boundary value problem for a nonlinear diffusion equation with a Volterra-type memory term in time is investigated. The model describes diffusion processes in media with hereditary (long-memory) effects and covers, as particular cases, a range of applications in heat and mass transfer. For its numerical solution we construct an implicit two-level difference scheme with a quadrature approximation of the memory integral. A discrete energy identity is derived, which yields energy stability of the scheme under natural constraints on the time step. The nonlinearity is treated by an inner iterative linearization procedure; at each iteration a tridiagonal system of linear algebraic equations arises and is solved efficiently by the sweep (Thomas) method. The convergence of the scheme is rigorously justified and its convergence rate is estimated in the corresponding energy norm. Numerical experiments are presented that confirm the theoretical error bounds and demonstrate the accurate reproduction of long-memory effects.