  We study the surface \bar{S} parametrizing cuboids: it is defined by the
equations relating the sides, face diagonals and long diagonal of a rectangular
box. It is an open problem whether a `rational box' exists, i.e., a rectangular
box all of whose sides, face diagonals and long diagonal have (positive)
rational length. The question is equivalent to the existence of nontrivial
rational points on \bar{S}.
  Let S be the minimal desingularization of \bar{S} (which has 48 isolated
singular points). The main result of this paper is the explicit determination
of the Picard group of S, including its structure as a Galois module over Q.
The main ingredient for showing that the known subgroup is actually the full
Picard group is the use of the combined action of the Galois group and the
geometric automorphism group of S (which we also determine) on the Picard
group. This reduces the proof to checking that the hyperplane section is not
divisible by 2 in the Picard group.
