The LHCb collaboration recently discovered a doubly charmed tetraquark $T_{cc}$ with flavor $cc\bar u\bar d$ just $0.36(4)~$MeV below $D^0D^{*+}$ threshold. This is the longest lived hadron with explicitly exotic quark content known to this date. We present the first lattice QCD study of $DD^*$ scattering in this channel, involving rigorous determination of pole singularities in the related scattering amplitudes that point to the existence of $T_{cc}$. Working with a heavier than physical light quark mass, we find evidence for a shallow virtual bound state pole in the $DD^*$ scattering amplitude with $l=0$, which is likely related to $T_{cc}$.