We introduce a new method for the formation of the J matrix (Coulomb interaction matrix) within a basis of Cartesian Gaussian functions, as needed in density functional theory and Hartree-Fock calculations. By summing the density matrix into the underlying Gaussian integral formulas, we have developed a J matrix "engine" which forms the exact J matrix without explicitly forming the full set of two electron integral intermediates. Several precomputable quantities have been identified, substantially reducing the number of floating point operations and memory accesses needed in a J matrix calculation. Initial timings indicate a speedup of greater than four times for the (pp/pp) class of integrals with speedups increasing to over ten times for (ff/ff) integrals.