The equivalent relation is established here about the stability of stochastic differential equations with piecewise continuous arguments(SDEPCAs) and that of the one-leg θ method applied to the SDEPCAs. Firstly, the convergence of the one-leg θ method to SDEPCAs under the global Lipschitz condition is proved. Secondly, it is proved that the SDEPCAs are pth(p ∈ (0, 1)) moment exponentially stable if and only if the one-leg θ method is pth moment exponentially stable for some sufficiently small step-size. Thirdly, the corollaries that the pth moment exponential stability of the SDEPCAs (the one-leg θ method) implies the almost sure exponential stability of the SDEPCAs (the one-leg θ method) are given. Finally, numerical simulations are provided to illustrate the theoretical results