  We prove that on any transitive graph $G$ with infinitely many ends, a
self-avoiding walk of length $n$ is ballistic with extremely high probability,
in the sense that there exist constants $c,t>0$ such that
$\mathbb{P}_n(d_G(w_0,w_n)\geq cn)\geq 1-e^{-tn}$ for every $n\geq 1$.
Furthermore, we show that the number of self-avoiding walks of length $n$ grows
asymptotically like $\mu_w^n$, in the sense that there exists $C>0$ such that
$\mu_w^n\leq c_n\leq C\mu_w^n$ for every $n\geq 1$. Our results extend more
generally to quasi-transitive graphs with infinitely many ends, satisfying the
additional technical property that there is a quasi-transitive group of
automorphisms of $G$ which does not fix an end of $G$.
