We study Kontsevich's deformation quantization for the dual of a finite-dimensional real Lie algebra (or superalgebra) g. In this case the Kontsevich star-product defines a new convolution on S(g), regarded as the space of distributions supported at 0 in g. For p in S(g), we show that the convolution operator f->f*p is a differential operator with analytic germ. We use this fact to prove a conjecture of Kashiwara and Vergne on invariant distributions on a Lie group. This yields a new proof of Duflo's result on local solvability of bi-invariant differential operators on a Lie group. Moreover, this new proof extends to Lie supergroups.