Compressive sensing (CS) shows that a sparse or compressible signal can be exactly recovered from its linear measurements at a rate significantly lower than the Nyquist rate. As an extreme case, 1-bit compressive sensing (1-bit CS) states that an original sparse signal can be recovered from the 1-bit measurements. In this paper, we intrduce a Fast and Accurate Two-Stage (FATS) algorithm for 1-bit compressive sensing. Simulations show that FATS not only significantly increases the signal reconstruction speed but also improves the reconstruction accuracy.