This paper gives a new criterion for input-output stability of a class of nonlinear feedback systems. Roughly speaking, it is most useful in such a practical situation where the nonlinearity in the system is "almost time-invariant and memoryless" but with "slight time-variations and dynamics." It involves two free parameters and contains the circle criterion and the Popov criterion as special cases. In fact, it extends these two famous criteria in such a way that the conservatism of the circle criterion can be reduced when the time-variations and dynamics of the nonlinearity are "relatively small." It is also shown that the existence of the free parameters that fulfill the stability condition can be checked exactly, by reducing it to a convex problem in the frequency domain.