  In this paper we construct a three-dimensional (3D) solvable lattice model
with non-negative Boltzmann weights. The spin variables in the model are
assigned to edges of the 3D cubic lattice and run over an infinite number of
discrete states. The Boltzmann weights satisfy the tetrahedron equation, which
is a 3D generalisation of the Yang-Baxter equation. The weights depend on a
free parameter 0<q<1 and three continuous field variables. The layer-to-layer
transfer matrices of the model form a two-parameter commutative family. This is
the first example of a solvable 3D lattice model with non-negative Boltzmann
weights.
