We deal with viscous perturbations of scalar conservation laws on a bounded interval with a general flux function f and a small dissipation coefficient e. Acting on this system on both endpoints of the interval, we prove global exact controllability to constant states with nonzero speed. More precisely, we construct boundary controls so that the solution is driven to the targeted constant state, and we moreover require these controls to be uniformly bounded as epsilon -> 0(+) in an appropriate space. For general (nonconvex) flux functions this can be done for sufficiently large time, and for convex fluxes f, we have a precise estimate on the minimal time needed to control.