Maybe you are asking for uniqueness, not existence? Then the answer is yes because your metric admits a locally CAT(0) completion in the answer above.

Maybe you are asking for uniqueness, not existence? Then the answer is yes because your metric admits a locally CAT(0) completion in the answer below. This uniqueness result, originally, I think, due to Teichmuller, should be also in Strebel's book and is based on a Gauss-Bonnet calculation.