  For locally finite CAT(0) cube complexes it is known that they are
injectively metrizable choosing the $l_\infty$-norm on each cube. In this paper
we show that cube complexes which are injective with respect to this metric are
always CAT(0). Moreover we give a criterion for finite dimensional CAT(0) cube
complexes with finite width to posses an injective metric. As a side result we
prove a modification of Bridson's Theorem for cube complexes saying that finite
dimensional cube complexes with $l_p$-norms on the cubes are geodesic.
