We report the discovery and analysis of a very strong magnetic field in the rapidly rotating early B-type star HR 5907, based on observations obtained as part of the Magnetism in Massive Stars (MiMeS) project. We infer a rotation period of 0.508 276+0.000 015 −0.000 012 d from photometric and Hα EW measurements, making this the shortest period, non-degenerate, magnetic massive star known to date. From the comparison of IUE UV and optical spectroscopy with LTE bruce/kylie models we find a solid-angle integrated, uniform black-body temperature of 17 000 ± 1000 K, a projected rotational velocity of 290 ± 10 km s−1, an equatorial radius of 3.1 ± 0.2 R⊙, a stellar mass of 5.5 ± 0.5 M⊙, and an inclination angle of the rotation axis to our line-of-sight of 70 ± 10°. Our measurements of the longitudinal magnetic field, which vary between −500 and −2000 G, phase coherently with the rotation period and imply a surface dipole field strength of ∼15.7 kG. On the other hand, from fits to mean Least-Squares Deconvolved Stokes V line profiles we infer a dipole field strength of ∼10.4 kG. This disagreement may result from a magnetic configuration more complex than our model, and/or from the non-uniform helium surface abundance distribution. In either case we obtain a magnetic obliquity nearly aligned with the rotation axis ( ). Our optical spectroscopy also shows weak variability in carbon, silicon and nitrogen lines. The emission variability in hydrogen Balmer and Paschen lines indicates the presence of a dense, highly structured magnetosphere, interpreted as a centrifugally supported, magnetically confined circumstellar disc.