We report the discovery of KELT-10b, the first transiting exoplanet discovered using the KELT-South telescope. KELT-10b is a highly inflated sub-Jupiter mass planet transiting a relatively bright $V = 10.7$ star (TYC 8378-64-1), with T$_{eff}$ = $5948pm74$ K, $log{g}$ = $4.319_{-0.030}^{+0.020}$ and [Fe/H] = $0.09_{-0.10}^{+0.11}$, an inferred mass M$_{*}$ = $1.112_{-0.061}^{+0.055}$ M$_{odot}$ and radius R$_{*}$ = $1.209_{-0.035}^{+0.047}$ R$_{odot}$. The planet has a radius R$_{P}$ = $1.399_{-0.049}^{+0.069}$ R$_{J}$ and mass M$_{P}$ = $0.679_{-0.038}^{+0.039}$ M$_{J}$. The planet has an eccentricity consistent with zero and a semi-major axis $a$ = $0.05250_{-0.00097}^{+0.00086}$ AU. The best fitting linear ephemeris is $T_{0}$ = 2457066.72045$pm$0.00027 BJD$_{TDB}$ and P = 4.1662739$pm$0.0000063 days. This planet joins a group of highly inflated transiting exoplanets with a radius much larger and a mass much less than those of Jupiter. The planet, which boasts deep transits of 1.4%, has a relatively high equilibrium temperature of T$_{eq}$ = $1377_{-23}^{+28}$ K, assuming zero albedo and perfect heat redistribution. KELT-10b receives an estimated insolation of $0.817_{-0.054}^{+0.068}$ $times$ 10$^9$ erg s$^{-1}$ cm$^{-2}$, which places it far above the insolation threshold above which hot Jupiters exhibit increasing amounts of radius inflation. Evolutionary analysis of the host star suggests that KELT-10b is unlikely to survive beyond the current subgiant phase, due to a concomitant in-spiral of the planet over the next $sim$1 Gyr. The planet transits a relatively bright star and exhibits the third largest transit depth of all transiting exoplanets with V $<$ 11 in the southern hemisphere, making it a promising candidate for future atmospheric characterization studies.
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