One of Ole Römer's most influential contributions to astronomy was the theory that light has a finite speed, which he calculated from inequalities in the motion of a satellite of Jupiter. The English astronomer John Flamsteed, first director of the Royal Observatory at Greenwich, is credited with being an early and influential supporter of the theory. This article examines how he came to be so, taking issue with the claim that he was instantly converted to the idea by a personal (...) encounter with Römer. The publication of Flamsteed's Correspondence has made illuminating material newly accessible; through letters it is possible to track the gradual development of his thinking and how it related to his dealings with Römer and with Giovanni Domenico Cassini (director of the Paris Observatory). The resulting account offers new insights into their attempts to share information, in a situation where motives for collaborating were counterbalanced by a potential for rivalry and conflict, and into the interlinked roles played by written correspondence, personal meetings and empirical investigation in the spreading of new ideas. (shrink)