  While cooler giant planets are often observed with non-zero eccentricities,
the short-period circular orbits of hot Jupiters suggest that they lose orbital
energy and angular momentum due to tidal interactions with their host stars.
However, orbital decay has never been unambiguously observed. We use data from
Gaia Data Release 2 to show that hot Jupiter host stars have a smaller Galactic
velocity dispersion than a similar population of stars without hot Jupiters.
Since Galactic velocity dispersion is correlated with age, this observation
implies that the population of hot Jupiter host stars is on average younger
than the field population. The best explanation for this inference is that
tidal interactions cause hot Jupiters to inspiral while their host stars are on
the main sequence. This observation requires that the typical modified stellar
tidal quality factor $Q_{\ast}^{'}$ for solar-type stars be in the range
$\log_{10}{Q_{\ast}^{'}} \lesssim 7$.
