The X-ray source M33 X-7 in the nearby galaxy Messier 33 is among the most massive X-ray binary stellar systems known, hosting a rapidly spinning, 15.65M(⊙) black hole orbiting an underluminous, 70M(⊙) main-sequence companion in a slightly eccentric 3.45-day orbit (M(⊙), solar mass). Although post-main-sequence mass transfer explains the masses and tight orbit, it leaves unexplained the observed X-ray luminosity, the star's underluminosity, the black hole's spin and the orbital eccentricity. A common envelope phase, or rotational mixing, could explain the orbit, but the former would lead to a merger and the latter to an overluminous companion. A merger would also ensue if mass transfer to the black hole were invoked for its spin-up. Here we report simulations of evolutionary tracks which reveal that if M33 X-7 started as a primary body of 85M(⊙)-99M(⊙) and a secondary body of 28M(⊙)-32M(⊙), in a 2.8-3.1-d orbit, its observed properties can be consistently explained. In this model, the main-sequence primary transfers part of its envelope to the secondary and loses the rest in a wind; it ends its life as a ∼16M(⊙) helium star with an iron-nickel core that collapses to a black hole (with or without an accompanying supernova). The release of binding energy, and possibly collapse asymmetries, 'kick' the nascent black hole into an eccentric orbit. Wind accretion explains the X-ray luminosity, and the black-hole spin can be natal.