The antiferromagnetic structure of the AF2 phase of Mn5Si3, stable between the Neel point at 99 K and the transition to a second antiferromagnetic AF1 phase at 66 K, has been determined from unpolarized neutron integrated intensity measurements on single-crystal and powder samples. The paramagnetic phase adopts the hexagonal D8(8) structure, Z = 2, with Mn1 and Mn2 atoms in a fourfold and a sixfold site respectively. With the onset of long-range magnetic order the symmetry is reduced to orthorhombic. The orthorhombic cell dimensions at 70 K are a = 6.898 56(1), b = 11.89120(2) and c = 4.793 30(1) Angstrom and these are related to those of the hexagonal cell, a(h) etc, by a approximate to a(h), b approximate to root 3a(h) and c approximate to c(h). The orthorhombic cell is C-face centred, space group C cmm, with Z = 4. Magnetic reflections indexed on this cell have h + k odd corresponding to a (010) magnetic propagation vector. Zero-field neutron polarimetry on a single crystal shows that there are no components of moment parallel to the c axis. A magnetic structure in which the Mn1 and one-third of the Mn2 atoms have no ordered moment and the remaining Mn2 atoms a moment of 1.48(1)mu(B) directed parallel and antiparallel to b gives a good fit to both the powder and single-crystal intensities. This structure differs significantly from previous models, but it is coherent with current ideas about the stability of Mn moments in compounds with B-subgroup elements. Its relation to the recently revised structure of the low-temperature AF1 phase is discussed.