We present dynamical measurements for 586 H{alpha}-detected star-forming galaxies from the KMOS (K-band Multi-Object Spectrograph) Redshift One Spectroscopic Survey (KROSS). The sample represents typical star-forming galaxies at this redshift (z=0.6-1.0), with a median star formation rate of ~=7M_{sun}_/yr and a stellar mass range of log(M*_[M_{sun}_])~9-11. We find that the rotation velocity-stellar mass relationship (the inverse of the Tully-Fisher relationship) for our rotationally dominated sources (v_C_/{sigma}_0_>1) has a consistent slope and normalization as that observed for z=0 discs. In contrast, the specific angular momentum (j*; angular momentum divided by stellar mass) is ~=0.2-0.3dex lower on average compared to z=0 discs. The specific angular momentum scales as j_s_{prop.to}M*^0.6+/-0.2^, consistent with that expected for dark matter (i.e. j_DM_{prop.to} M_DM_^2/3^). We find that z~=0.9 star-forming galaxies have decreasing specific angular momentum with increasing Sersic index. Visually, the sources with the highest specific angular momentum, for a given mass, have the most disc-dominated morphologies. This implies that an angular momentum-mass-morphology relationship, similar to that observed in local massive galaxies, is already in place by z~=1.