We consider a small metallic island biased by an external voltage source via a tunnel junction. The dependence of the average charge in this so-called single electron box on the bias voltage displays steps representing a prototypical Coulomb staircase. The steps are smeared by quantum fluctuations arising from the finite tunneling conductance of the junction. The rounding of the stairs is determined for the realistic case where the electronic bandwidth is large compared to the single electron charging energy.