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 ~7 Msol/yr and a stellar mass range of log[M/Msol]~9-11. We find that the rotation velocity-stellar mass relationship (the inverse of the Tully-Fisher relationship) for our rotationally-dominated sources (v/sigma>1) has a consistent slope and normalisation as that observed for z=0 disks. In contrast, the specific angular momentum (j; angular momentum divided by stellar mass), is ~0.2-0.3 dex lower on average compared to z=0 disks. The specific angular momentum scales as M^[0.6+/-0.2], consistent with that expected for dark matter (i.e., proportional to M^[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 disk-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. ; Comment: Accepted for publication in MNRAS. Table of values and extended figures available at: http://astro.dur.ac.uk/KROSS/