An experimental investigation is made of the three-dimensional boundary layer that results when a Rankine-like vortex is bounded by a fixed plane boundary, in particular by a horizontal disc coaxial with, and perpendicular to, the axis of rotation of the vortex. A laser-Doppler anemometer is used to make velocity traverses through both the vortex and the boundary layer, for Reynolds numbers, Re, ranging from 5000 to 30 000, where Re is based on velocity and radius at the disc edge. The boundary layer is laminar at Re = 5000 and the data agree well with the theory of Belcher et al. (J. Fluid Mech. 52, 753-780 (1972)); at Re = 10 000 the layer is in a transitory state, while for Re $\geqslant$ 15 000 it is turbulent over some of the disc. The radial pressure gradient associated with the outer flow has a stabilizing effect on the boundary layer and, for 10 000 $\leqslant$ Re $\leqslant$ 30 000, acts to revert it to a laminar state, but with diminishing effect as Re increases. In spite of the high three-dimensionality of the layer, the tangential component of velocity conforms to the same law-of-the-wall as its streamwise counterpart in two-dimensional turbulent boundary layers.