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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.