The injection of a viscous fluid into a mould formed by two parallel plates is considered. The flow front is supposed to move at constant speed. It is assumed that there is complete adherence between the fluid and the mould walls, and that the environmental pressure is constant. For a Newtonian fluid the problem is described in terms of two analytic complex functions. The shape of the fluid surface is calculated by means of a conformal-mapping technique, which leads to a Hilbert problem. The results are compared with known finite-element simulations.