Kepler 1627A is a G8V star previously known to host a 3.8 R ⊕ planet on a 7.2 day orbit. The star was observed by the Kepler space telescope because it is nearby (d = 329 pc) and it resembles the Sun. Here, we show using Gaia kinematics, TESS stellar rotation periods, and spectroscopic lithium abundances that Kepler 1627 is a member of the ${38}_{-5}^{+6}$ Myr old δ Lyr cluster. To our knowledge, this makes Kepler 1627Ab the youngest planet with a precise age yet found by the prime Kepler mission. The Kepler photometry shows two peculiarities: the average transit profile is asymmetric, and the individual transit times might be correlated with the local light-curve slope. We discuss possible explanations for each anomaly. More importantly, the δ Lyr cluster is one of ~103 coeval groups whose properties have been clarified by Gaia. Many other exoplanet hosts are candidate members of these clusters; their ages can be verified with the trifecta of Gaia, TESS, and ground-based spectroscopy.