We have performed two-photon ultraviolet direct frequency comb spectroscopy on the 1S-3S transition in atomic hydrogen to illuminate the so-called proton radius puzzle and to demonstrate the potential of this method. The proton radius puzzle is a significant discrepancy between data obtained with muonic hydrogen and regular atomic hydrogen that could not be explained within the framework of quantum electrodynamics. By combining our result [f(1s-3s) = 2,922,743,278,665.79(72) kilohertz] with a previous measurement of the 1S-2S transition frequency, we obtained new values for the Rydberg constant [R-infinity = 10,973,731.568226(38) per meter] and the proton charge radius [r(p) = 0.8482(38) femtometers]. This result favors the muonic value over the world-average data as presented by the most recent published CODATA 2014 adjustment.