We report experiments on the rapid contact-line motion present in the early stages of capillary-driven spreading of drops on dry solid substrates. The spreading data fail to follow a conventional viscous or inertial scaling. By integrating experiments and simulations, we quantify a contact-line friction μ(f) which is seen to limit the speed of the rapid dynamic wetting. A scaling based on this contact-line friction is shown to yield a universal curve for the evolution of the contact-line radius as a function of time, for a range of fluid viscosities, drop sizes, and surface wettabilities.