  In this paper, we introduce a new discretization of the Gaussian curvature on
surfaces, which is defined as the quotient of the angle defect and the area of
some dual cell of a weighted triangulation at the conic singularity. A discrete
uniformization theorem for this discrete Gaussian curvature is established on
surfaces with non-positive Euler number. The main tools are Bobenko-Lutz's
discrete conformal theory for decorated piecewise Euclidean metrics on surfaces
and variational principles with constraints.
