This paper describes a block (affine) projection algorithm that has exactly the same convergence rate as the original sample-by-sample algorithm and smaller computational complexity than the fast affine projection algorithm. This is achieved by (1) introducing a correction term that compensates for the filter output difference between the sample-by-sample projection algorithm and the straightforward block projection algorithm, and (2) applying a fast finite impulse response (FIR) filtering technique to compute the filter outputs and to update the filter. We describe how to choose a pair of block lengths that gives the longest filter length under a constraint on the total computational complexity and processing delay. An example shows that the filter length can be doubled if a delay of a few hundred samples is permissible.