Uniform embeddability and exactness of free products

Let A and B be countable discrete groups and let Γ=A∗B be their free product. We show that if both A and B are uniformly embeddable in a Hilbert space then so is Γ. We give two different proofs: the first directly constructs a uniform embedding of Γ from uniform embeddings of A and B; the second works without change to show that if both A and B are exact then so is Γ.