Let $M$ be a closed, connected, smooth three-manifold with $\pi_1(M) = 0$, and let $g_0$ be a smooth Riemannian metric on $M$. Let $(M(t), g(t))$ be a three-dimensional Ricci flow with surgery starting from $(M, g_0)$, constructed using Perelman's surgery parameters and satisfying the canonical-neighbourhood assumptions, Hamilton-Ivey pinching estimates, and $\kappa$-noncollapsing controls required in Perelman's surgery construction. Then the surgically modified flow becomes extinct in finite time: there exists $T < \infty$ such that $M(t) = \varnothing$ for every surgery time or regular time $t \geq T$.