  Charge density waves are emergent quantum states that spontaneously reduce
crystal symmetry, drive metal-insulator transitions, and precede
superconductivity. In low-dimensions, distinct quantum states arise, however,
thermal fluctuations and external disorder destroy long-range order. Here we
stabilize ordered two-dimensional (2D) charge density waves through endotaxial
synthesis of confined monolayers of 1T-TaS$_2$. Specifically, an ordered
incommensurate charge density wave (oIC-CDW) is realized in 2D with
dramatically enhanced amplitude and resistivity. By enhancing CDW order, the
hexatic nature of charge density waves becomes observable. Upon heating via
in-situ TEM, the CDW continuously melts in a reversible hexatic process wherein
topological defects form in the charge density wave. From these results, new
regimes of the CDW phase diagram for 1T-TaS$_2$ are derived and consistent with
the predicted emergence of vestigial quantum order.
