The momentum correction coefficient in one-dimensional blood flow models is related to the prescribed velocity profile. The exact solution of the Riemann problem for the one-dimensional blood flow equations has been previously studied only for a momentum correction coefficient equal to one (corresponding to a flat velocity profile, i.e. an inviscid fluid). In this paper we solve exactly the Riemann problem for the non-linear hyperbolic one-dimensional blood flow equations with a general constant momentum correction coefficient and a tube law that allows to describe both arteries and veins with continuous or discontinuous mechanical and geometrical properties. In the case of discontinuous properties, only the subsonic regime is considered. We propose a numerical procedure to compute the obtained exact solution and finally we validate it numerically, by comparing exact solutions to those obtained with well-known, first order, numerical schemes on a carefully designed set of test problems. A detailed knowledge about this problem will allow to determine coupling and boundary conditions arising when these models are applied on networks of vessels, ensuring full consistency with the underlying one-dimensional blood flow model without resorting to linearization techniques commonly applied when the momentum correction coefficient is assumed to be different from one.