The extreme astrophysical processes and conditions that characterize the early Universe are expected to result in young galaxies that are dynamically different from those observed today(1-5). This is because the strong effects associated with galaxy mergers and supernova explosions would lead to most young star-forming galaxies being dynamically hot, chaotic and strongly unstable(1,2). Here we report the presence of a dynamically cold, but highly star-forming, rotating disk in a galaxy at redshift(6)z = 4.2, when the Universe was just 1.4 billion years old. Galaxy SPT-S J041839-4751.9 is strongly gravitationally lensed by a foreground galaxy atz = 0.263, and it is a typical dusty starburst, with global star-forming(7)and dust properties(8)that are in agreement with current numerical simulations(9)and observations(10). Interferometric imaging at a spatial resolution of about 60 parsecs reveals a ratio of rotational to random motions of 9.7 +/- 0.4, which is at least four times larger than that expected from any galaxy evolution model at this epoch(1-5)but similar to the ratios of spiral galaxies in the local Universe(11). We derive a rotation curve with the typical shape of nearby massive spiral galaxies, which demonstrates that at least some young galaxies are dynamically akin to those observed in the local Universe, and only weakly affected by extreme physical processes. A strongly lensed galaxy at redshift 4.2 appears to be a dynamically cold disk galaxy, similar to spiral galaxies in the local neighbourhood and weakly affected by extreme physical processes.