The Colonel Blotto game is a famous game commonly used to model resource allocation problems in domains ranging from security to advertising. Two players distribute a fixed budget of resources on multiple battlefields to maximize the aggregate value of battlefields they win, each battlefield being won by the player who allocates more resources to it. Recently, the discrete version of the game-where allocations can only be integers-started to gain traction and algorithms were proposed to compute the equilibrium in polynomial time; but these remain computationally impractical for large (or even moderate) numbers of battlefields. In this paper, we propose an algorithm to compute very efficiently an approximate equilibrium for the discrete Colonel Blotto game with many battlefields. We provide a theoretical bound on the approximation error as a function of the game's parameters. Through numerical experiments, we show that the proposed strategy provides a fast and good approximation even for moderate numbers of battlefields.