Prime II1 factors arising from irreducible lattices in products of rank one simple Lie groups

We prove that if Gamma is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the II1 factor L(Gamma) is prime. In particular, we deduce that the II1 factors associated to the arithmetic groups PSL2(Z[root d]) and PSL2(Z[S-1]) are prime for any square-free integer d >= 2 with d not equivalent to 1 (mod 4) and any finite non-empty set of primes S. This provides the first examples of prime II1 factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe all tensor product decompositions of L(Gamma) for icc countable groups Gamma that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that L(Gamma) is prime, unless Gamma is a product of infinite groups, in which case we prove a unique prime factorization result for L(Gamma).