We report the discovery of a new ultra-short period transiting hot Jupiter from the Next Generation Transit Survey (NGTS). NGTS-10b has a mass and radius of $2.162,^{+0.092}_{-0.107}$ M$_{rm J}$ and $1.205,^{+0.117}_{-0.083}$ R$_{rm J}$ and orbits its host star with a period of $0.7668944pm0.0000003$ days, making it the shortest period hot Jupiter yet discovered. The host is a $10.4pm2.5$ Gyr old K5V star ($T_mathrm{eff}$=$4400pm100$,K) of Solar metallicity ([Fe/H] = $-0.02pm0.12$,dex) showing moderate signs of stellar activity. NGTS-10b joins a short list of ultra-short period Jupiters that are prime candidates for the study of star-planet tidal interactions. NGTS-10b orbits its host at just $1.46pm0.18$ Roche radii, and we calculate a median remaining inspiral time of $38$,Myr and a potentially measurable transit time shift of $7$,seconds over the coming decade, assuming a stellar tidal quality factor $Q_{rm s}=2times10^{7}$.
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