We present a numerical integration scheme designed to treat the type of multicenter integrals encountered in electronic structure calculations. By developing a notation that differentiates between those atomic centers where integrands have significant amplitudes and those where they do not, we find a way to decompose multicenter integrals (into sums over one-center integrals) such that the number of operations needed for a given matrix element does not increase with increasing system size. In addition, a new adaptive one-center grid is presented that accounts for the shell structures of core electrons while allowing for the vastly different behavior of integrands in the valence and tail regions. Through the use of model integrands the necessary grid points are automatically generated for a given system based on the accuracy requested. Our new multicenter decomposition scheme and one-center grid have been tested separately and in conjunction with each other. Results of such tests demonstrate that our decomposition scheme combined with our one-center grid provides significant improvements over existing multicenter integration schemes. In addition to demonstrating the efficiency of the method for any size system, we will show that the CPU cost of an integral remains constant for systems larger than some easily achievable threshold size. In general comparison shows that the larger the system, the higher is the percent gain in efficiency over previously published methods. In addition, the higher the accuracy targeted, the higher percentage the gain. Also, the higher the accuracy required for a given system, the higher is the gain ire efficiency. The method is therefore of great use for large polyatomic molecules and periodic systems.