Summary: We deal with the viscous Burgers equation with a small dissipation coefficient \(\nu\). We prove the (global) exact controllability property to nonzero constant states, that is to say, the possibility of finding boundary values such that the solution of the associated Burgers equation is driven to a constant state. The main objective of this paper is to do so with control functions whose norms in an appropriate space are bounded independently of \(\nu\), which belongs to a suitably small interval. This result is obtained for a sufficiently large time.