The laser and the transistor are two of the most important inventions of the 20th century. Basic agreement exists that Bardeen, Brattain, and Shockley invented the transistor. Inventorship priority over the laser has been contested. By mobilizing patent and patent-related sources, traditional scientific literature, published commentaries, interviews with several of the surviving principals, and the Townes papers in the Library of Congress, we are able to suggest an invention priority for the laser that we believe improves on previous scholarship. We conclude that there is no single inventor of the laser, but that the most fundamental U.S. contributions were made by Townes, Schawlow, and Bloembergen. We believe the logic used here for deciding invention priority, which includes a careful study of patent claims, may be useful in other cases where inventorship is of interest.