We consider stochastic differential equations dY=V(Y)dX driven by a multidimensional Gaussian process X in the rough path sense [T. Lyons, Rev. Mat. Iberoamericana 14, (1998), 215--310]. Using Malliavin Calculus we show that Yt admits a density for t∈(0,T] provided (i) the vector fields V=(V1,…,Vd) satisfy Hörmander’s condition and (ii) the Gaussian driving signal X satisfies certain conditions. Examples of driving signals include fractional Brownian motion with Hurst parameter H>1/4, the Brownian bridge returning to zero after time T and the Ornstein-Uhlenbeck process.