In 1985, Kulkarni defined the conformal boundary ∂ℒ of a simply connected and time-oriented Lorentzian surface ℒ. He also introduced a notion of 'smoothability' of this boundary, depending only on local properties of ∂ℒ. In this paper we show that smoothability of ∂ℒ is in fact a global property of ℒ. In doing so, we classify Lorentzian surfaces with smoothable boundaries up to conformal homeomorphism. To be specific, suppose that the Minkowski plane E21 is the x,y-plane with metric dxdy. Our main theorem states that if ∂ℒ is smoothable then ℒ is conformally homeomorphic to the interior U of a Jordan curve in E21 that is locally the graph of a continuous function over either the x-axis or the y-axis at each point of ∂U.