We have investigated the start-up flow and yielding of a simple yield-stress fluid (Carbopol 940) in a vertical pipe. The Carbopol was displaced from below by a Newtonian liquid (Fluorinert FC-40) injected at the bottom of the pipe. Rough and smooth-walled pipes were used to study the effects of wall boundary conditions. The shear stress at the wall was known from the pressure in the Carbopol measured by a pressure gauge fixed on the pipe wall. The shear rate was known from measurements of velocity profile in the Carbopol made using particle image velocimetry. In the rough-walled pipe, the yielding involved a long transit with several steps: elastic deformation, the onset of wall slip, yielding at the wall, and finally a steady-state plug flow that is well-described by the predictions of the Herschel-Bulkley model. In contrast, in the smooth-walled pipe, the wall shear stress never exceeded the yield stress. Our results demonstrate the importance of elasticity during the yielding process, particularly for faster displacements.