  The equations for a G_2-structure with torsion on a product M^7 = N^6 x S^1
are studied in relation to the induced SU(3)-structure on N^6. All solutions
are found in the case when the Lee-form of the G_2-structure is non-zero and
N^6 is a six-dimensional nilmanifold with half-integrable SU(3)-structure.
Special properties of the torsion of these solutions are discussed.
