We prove that the Opdam-Cherednik kernel, also known as the nonsymmetric Opdam hypergeometric function, can be written as the Laplace transform of a positive measure supported on the convex hull of the Weyl group orbit of its argument. As a consequence, the trigonometric Dunkl intertwining operator is positivity preserving. The main ingredient in the proof is a new formula for the Opdam-Cherednik kernel as a degeneration of nonsymmetric Macdonald polynomials. As a further application, we prove majorization inequalities for Macdonald polynomials and Heckman-Opdam hypergeometric functions associated with arbitrary root systems.