Let $(M,g(t))$ be a smooth Ricci flow on a closed $n$-manifold for $t \in [0,T)$. For every $T_0<T$ there exist constants $\kappa>0$ and $r_0>0$, depending on the initial metric and $T_0$, such that for every $t \in [0,T_0]$, every $x \in M$, and every $0<r\le r_0$, if $|\operatorname{Rm}_{g(t)}|\le r^{-2}$ on the geodesic ball $B_{g(t)}(x,r)$, then $\operatorname{Vol}_{g(t)}(B_{g(t)}(x,r))\ge \kappa r^n$.