We report the discovery of KELT-12b, a highly inflated Jupiter-mass planet transiting a mildly evolved host star. We identified the initial transit signal in the KELT-North survey data and established the planetary nature of the companion through precise follow-up photometry, high-resolution spectroscopy, precise radial velocity measurements, and high-resolution adaptive optics imaging. Our preferred best-fit model indicates that the $V = 10.64$ host, TYC 2619-1057-1, has $T_{\rm eff} = 6278 \pm 51$ K, $\log{g_\star} = 3.89^{+0.054}_{-0.051}$, and Fe/H = $0.19^{+0.083}_{-0.085}$, with an inferred mass $M_{\star} = 1.59^{+0.071}_{-0.091} M_\odot$ and radius $R_\star = 2.37 \pm 0.18 R_\odot$. The planetary companion has $M_{\rm P} = 0.95 \pm 0.14 M_{\rm J}$, $R_{\rm P} = 1.79^{+0.18}_{-0.17} R_{\rm J}$, $\log{g_{\rm P}} = 2.87^{+0.097}_{-0.098}$, and density $\rho_{\rm P} = 0.21^{+0.075}_{-0.054}$ g cm$^{-3}$, making it one of the most inflated giant planets known. The time of inferior conjunction in ${\rm BJD_{TDB}}$ is $2457088.692055 \pm 0.0009$ and the period is $P = 5.0316144 \pm 0.0000306$ days. Despite the relatively large separation of $\sim0.07$ AU implied by its $\sim 5.03$-day orbital period, KELT-12b receives significant flux of $2.93^{+0.33}_{-0.30} \times 10^9$ erg s$^{-1}$ cm$^{-2}$ from its host. We compare the radii and insolations of transiting gas-giant planets around hot ($T_{\rm eff} \geq 6250$ K) and cool stars, noting that the observed paucity of known transiting giants around hot stars with low insolation is likely due to selection effects. We underscore the significance of long-term ground-based monitoring of hot stars and space-based targeting of hot stars with the Transiting Exoplanet Survey Satellite (TESS) to search for inflated giants in longer-period orbits.