  The Transiting Exoplanet Survey Satellite mission was designed to find
transiting planets around bright, nearby stars. Here we present the detection
and mass measurement of a small, short-period ($\approx\,4$\,days) transiting
planet around the bright ($V=7.9$), solar-type star HD 86226 (TOI-652, TIC
22221375), previously known to host a long-period ($\sim$1600 days) giant
planet. HD 86226c (TOI-652.01) has a radius of $2.16\pm0.08$ $R_{\oplus}$ and a
mass of 7.25$^{+1.19}_{-1.12}$ $M_{\oplus}$ based on archival and new radial
velocity data. We also update the parameters of the longer-period,
not-known-to-transit planet, and find it to be less eccentric and less massive
than previously reported. The density of the transiting planet is $3.97$ g
cm$^{-3}$, which is low enough to suggest that the planet has at least a small
volatile envelope, but the mass fractions of rock, iron, and water are not
well-constrained. Given the host star brightness, planet period, and location
of the planet near both the ``radius gap'' and the ``hot Neptune desert'', HD
86226c is an interesting candidate for transmission spectroscopy to further
refine its composition.

Earth and Planetary Astrophysics