We derive a CUR approximate matrix factorization based on the discrete empirical interpolation method (DEIM). For a given matrix A, such a factorization provides a low-rank approximate decomposition of the form A≈CUR, where C and R are subsets of the columns and rows of A, and U is constructed to make $\bf C\bf U \bf R $ a good approximation. Given a low-rank singular value decomposition A≈VSWT, the DEIM procedure uses V and W to select the columns and rows of A that form C and R. Through an error analysis applicable to a general class of CUR factorizations, we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of V and W. For very large problems, V and W can be approximated well using an incremental QR algorithm that makes only one pass through A. Numerical examples illustrate the favorable performance of the DEIM-CUR method compared to CUR approximations based on leverage scores.