The descent algebra Sigma(W) is a subalgebra of the group algebra QW of a finite Coxeter group W, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of W. Thus Sigma(W) is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct Sigma(W) as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of S, the set of simple reflections in W. From this construction we obtain some general information about the quiver of Sigma(W) and an algorithm for the construction of a quiver presentation for the descent algebra Sigma(W) of any given finite Coxeter group W. (C) 2008 Elsevier Inc. All rights reserved.