A fast algorithm for the approximate multiplication of matrices with decay is introduced; the Sparse Approximate Matrix Multiply (SpAMM) reduces complexity in the product space, a different approach from current methods that economize within the matrix space through truncation or rank reduction. Matrix truncation (element dropping) is compared to SpAMM for quantum chemical matrices with approximate exponential and algebraic decay. For matched errors in the electronic total energy, SpAMM is found to require fewer to far fewer floating point operations relative to dropping. The challenges and opportunities afforded by this new approach are discussed, including the potential for high performance implementations.