The theorem you stated can be true only for genus zero (that is for the sphere), if $K(x)<0$ at some point $x$); this follows from the Gauss Bonnet theorem that integral of the curvature $-K$ is equal to the Euler characteristic. 
It is indeed true, and  follows from the results of Troyanov, Prescribing curvature on compact surfaces with conical singularities.
Trans. Amer. Math. Soc. 324 (1991), no. 2, 793–821.

To obtain such metrics on surfaces of higher genus, one has to permit
conic singularities of the metric. This is a hot research topic nowadays, and there is a <a href="https://www.math.purdue.edu/~eremenko/dvi/survey.pdf">recent survey</a>.