An algorithm is given that decodes the Leech lattice with not much more than twice the complexity of soft-decision decoding of the Golay code. The algorithm has the same effective minimum distance as maximum-likelihood decoding and increases the effective error coefficient by less than a factor or two. The algorithm can be recognized as a member of the class of multistage algorithms that are applicable to hierarchical constructions. It is readily generalized to lattices that can be expressed in terms of binary code formulas, and in particular to construction B lattices. >