A standard expectation of primordial cosmological inflation is that it dilutes all relics created before its onset to unobservable levels. We present a counterexample to this expectation by demonstrating that a network of cosmic strings diluted by inflation can regrow to a level that is potentially observable today in gravitational waves (GWs). In contrast to undiluted cosmic strings, whose primary GW signals are typically in the form of a stochastic GW background, the leading signal from a diluted cosmic string network can be distinctive bursts of GWs within the sensitivity reach of current and future GW observatories.