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An n-simplex in H n with vertices v0, ..., v , 6 Hn U ~H ~ is the dosed subset of Hn bounded by the n + I spheres which contain all the vertices except one and which are orthogonal to S "-1. k simplex is called ideal if all the vertices arc on the sphere at infinity. I t is easy to see tha t the volume of a hyperbolic n-simplex is finite also ff some of the vertices are on the sphere at infinity. A simplex is called regular if any permuta t ion of its vertices can be induced by an isometry of H n. This makes sense also for ideal simplices since any i sometry of H n can be extended continuously to HnU OH n. There is, up to isometry, only one ideal regular n-simplex in H ~. The main result of the present paper is the following theorem which was conjectured by Thurs ton ([6], section 6.1).

Simplices of maximal volume in hyperbolic n-space