By numerically computing the steady axisymmetric flow of helium II confined inside a finite-aspect-ratio Couette annulus, we determine the transition from Ekman flow to Taylor vortex flow as a function of temperature and aspect ratio. We find that the low-Reynolds-number flow is quite different to that of a classical fluid, particularly at lower temperatures. At high aspect ratio our results confirm the existing linear stability theory of the onset of Taylor vortices, which assumes infinitely long cylinders.