In this work, we study priori error estimates for the numerical approximation of an optimal control problem governed by the heat equation with certain control constraint and ending point state constraint. By making use of the classical space-time discretization scheme, namely, finite element method with the space variable and backward Euler discretization for the time variable, we first project the original optimal control problem into a semi-discrete control and state constrained optimal control problem governed by an ordinary differential equation, and then project the aforementioned semi-discrete problem into a fully discrete optimization problem with constraints. With the help of Pontryagin's maximum principle, we obtain, under a certain reasonable condition of Slater style, not only an error estimate between the optimal controls for the original problem and the semi-discrete problem, but also an error estimate between the solutions of the semi-discrete problem and the fully discrete problem, which leads to an error estimate between the solutions of the original problem and the fully discrete problem. By making use of the aforementioned result, we also establish an numerical approximation for the exactly null controllability of the internally controlled heat equation.