  We provide the first mathematical proof that the connective constant of the
hexagonal lattice is equal to $\sqrt{2+\sqrt 2}$. This value has been derived
non rigorously by B. Nienhuis in 1982, using Coulomb gas approach from
theoretical physics. Our proof uses a parafermionic observable for the self
avoiding walk, which satisfies a half of the discrete Cauchy-Riemann relations.
Establishing the other half of the relations (which conjecturally holds in the
scaling limit) would also imply convergence of the self-avoiding walk to
SLE(8/3).
