We describe a hidden mechanism for stable mode locking of solid-state lasers that uses a slow saturable absorber. The physical origin of this novel mechanism is attributed to the transformation of the saturable-absorber-induced self-frequency shift into a fast nonlinear loss because the absorption line is off resonance. Based on the exact solution of the governing steady-state equation, we investigate in detail the dependence of the pulse parameters on the laser control parameters in connection with the stability region of the solution.