Gravitational waves (GWs) produce small distortions in the observable distribution of stars in the sky. We describe the characteristic pattern of astrometric deflections created by a specific gravitational waveform called a burst with memory. Memory is a permanent, residual distortion of space left in the wake of GWs. We demonstrate that the astrometric effects of GW memory are qualitatively distinct from those of more broadly considered, oscillatory GWs---distinct in ways with potentially far-reaching observational implications. We discuss some such implications pertaining to the random-walk development of memory-induced deflection signatures over cosmological time spans and how those may influence observations of the cosmic microwave background.