  We prove that Wang's free orthogonal and free unitary quantum groups are
weakly amenable and that their Cowling-Haagerup constant is equal to 1. This is
achieved by estimating the completely bounded norm of the projections on the
coefficients of irreducible representations of their compact duals. An argument
of monoidal equivalence then allows us to extend this result to quantum
automorphism groups of finite spaces and even yields some examples of weakly
amenable non-unimodular discrete quantum groups with the Haagerup property.
