The discovery of graphene some ten years ago was the first proof of a free-standing two-dimensional (2D) solid phase. Here, using quantum molecular dynamics simulations of nanoscale gold patches suspended in graphene pores, we predict the existence of an atomically thin, free-standing 2D liquid phase. The liquid phase, enabled by the exceptional planar stability of gold due to relativistic effects, demonstrates extreme fluxionality of metal nanostructures and opens possibilities for a variety of nanoscale phenomena.