We prove that any projective Schur algebra over a field K is equivalent in Br(K) to a radical abelian algebra. This was conjectured in 1995 by Sonn and the first author of this paper. As a consequence, we obtain a characterization of the projective Schur group by means of Galois cohomology. The conjecture was known for algebras over fields of positive characteristic. In characteristic zero the conjecture was known for algebras over fields with a Henselian valuation over a local or global field of characteristic zero.