We present the discovery of NGTS-1b, a hot-Jupiter transiting an early M-dwarf host ($Teff=3916-63+71 K$) in a P=2.674d orbit discovered as part of the Next Generation Transit Survey (NGTS). The planet has a mass of $0.812-0.075+0.066 MJ$, making it the most massive planet ever discovered transiting an M-dwarf. The radius of the planet is $1.33-0.33+0.61 RJ$. Since the transit is grazing, we determine this radius by modelling the data and placing a prior on the density from the population of known gas giant planets. NGTS-1b is the third transiting giant planet found around an M-dwarf, reinforcing the notion that close-in gas giants can form and migrate similar to the known population of hot Jupiters around solar type stars. The host star shows no signs of activity, and the kinematics hint at the star being from the thick disk population. With a deep (2.5%) transit around a $K=11.9$ host, NGTS-1b will be a strong candidate to probe giant planet composition around M-dwarfs via JWST transmission spectroscopy.