This paper presents a detailed theoretical analysis of the Langevin Monte Carlo sampling algorithm recently introduced in [DMP16] when applied to log-concave probability distributions that are restricted to a convex body K. This method relies on a regularisation procedure involving the Moreau-Yosida envelope of the indicator function associated with K. Explicit convergence bounds in total variation norm and in Wasserstein distance of order 1 are established. In particular, we show that the complexity of this algorithm given a first order oracle is polynomial in the dimension of the state space. Finally, some numerical experiments are presented to compare our method with competing MCMC approaches from the literature.