Results. We obtain a spectroscopic mass ratio q = 0.31 +/- 0.03 together with donor and gainer masses M-d = 2.2 +/- 0.2 M-circle dot, M-g = 7.3 +/- 0.3 M-circle dot, the radii R-d = 8.4 +/- 0.2 R-circle dot, R-g = 3.6 +/- 0.2 R-circle dot and temperatures T-d = 9400 +/- 100 K, T-g = 18 500 +/- 500 K for the stellar components. We find that DQ Vel is a semi-detached system consisting of a B3V gainer and an A1III donor star plus an extended accretion disc around the gainer. The disc is filling 89% of the gainer Roche lobe with a temperature of 6580 +/- 300 K at the outer radius. It has a concave shape that is thicker at its edge (d(e) = 0.6 +/- 0.1 R-circle dot) than at its centre (d(c) = 0.3 +/- 0.1 R-circle dot). We find a significant sub-orbital frequency of 0.19 d(-1) in the residuals of the V-band light curve, which we interpret as a pulsation of an slowly pulsating B-type (SPB) of a gainer star. We also estimate the distance to the binary (d similar to 3.1 kpc) using the absolute radii, apparent magnitudes, and effective temperatures of the components found in our study.