The moduli space of cubic threefolds in $mathbb{C}P^4$, with some minor birational modifications, is the Baily-Borel compactification of the quotient of the complex 10-ball by a discrete group. The authors describe both the birational modifications and the discrete group explicitly.|The moduli space of cubic threefolds in $mathbb{C}P^4$, with some minor birational modifications, is the Baily-Borel compactification of the quotient of the complex 10-ball by a discrete group. The authors describe both the birational modifications and the discrete group explicitly.