The crystal and magnetic structures of alpha-NiMoO4 were determined by single-crystal X-ray diffraction on the one hand, and elastic neutron powder scattering on the other. The compound was prepared by subsolidus reaction and crystallizes with Ni and Mo ions in oxygen octahedra, space group C2/m. Below 19 K antiferromagnetic ordering is observed. The magnetic unit cell has a doubled c-axis, i.e. k = (0,0,1/2), and the magnetic space group is C(2C)2/m'. The Ni ions occupy two different sites, so that two sublattices have to be distinguished. By neglecting the small canting angle between these sublattices, the magnetic structure is collinear and all moments within one chemical cell are parallel. The magnetic structure can be described by a sequence of such cells with alternating signs along the c-direction. At 1.5 K the absolute values of the magnetic moments and their canting angles with the alpha-axis (in the +alpha,+ c quadrant) are determined to be \mu(Ni2+)\ = 1.62(10)mu(B) and 75(10)degrees for the ions on mirror planes (Ni(1) site) and to \mu(Ni2+)\ = 1.45(15)mu(B) and 55(10)degrees for the ions on screw axes (Ni(2) site). From a Curie-Weiss fit in the paramagnetic region a Curie-Weiss temperature of 22(2)K and a magnetic moment of 3.05(5)mu(B) for each Ni2+ ion are derived. These values are in good agreement with those predicted by Hund's rules.