We describe with a probabilistic viewpoint the convolution product of two conjugacy classes of the unitary group 푈(푛). The description is given in terms of a probability distribution on the space of central measures which admits a density. Relating the convolution to the quantum Littlewood-Richardson coefficients and using recent results describing those coefficients, we give a positive formula for this density. In the same flavor as the hive model of Knutson and Tao, this formula is given in terms of a subtraction-free sum of volumes of explicit polytopes.