The Heston stochastic volatility model is one of the most fundamental models in mathematical finance. In the literature, numerous efficient simulation methods have been proposed. However, there is no analytical weak convergence rate that applies to the full parameter regime, which is a long-standing problem. In this paper, we derive the weak convergence rate of a time-discrete scheme for the Heston stochastic volatility model, which employs the stochastic trapezoidal rule to discretize the logarithmic asset process, provided that the variance process is simulated exactly. The test function we consider for the error criterion can be any polynomials of the logarithmic asset process. We show that the analytical weak convergence rate is 2 for all parameter regimes. The result is consistent with the standard rate of the stochastic trapezoidal rule, although the coefficient of the model does not satisfy the global Lipschitz condition. Furthermore, in the majority of the literature, the analysis is based on assumptions on coefficients of SDEs, whereas we impose no direct conditions on the coefficient. Instead, we transfer the error analysis to that of a trapezoidal rule on a deterministic integral, on which we make some mild assumptions. Finally, we extend the scheme and the analysis for more general SDEs.