  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.
