We use the new ultra-deep , near-infrared imaging of the Hubble Ultra-Deep Field ( HUDF ) provided by our UDF12 Hubble Space Telescope ( HST ) WFC3/IR campaign to explore the rest-frame ultraviolet ( UV ) properties of galaxies at redshifts z > 6.5 . We present the first unbiased measurement of the average UV power-law index , \langle \beta \rangle , ( f _ { \lambda } ~ { } \propto~ { } \lambda ^ { \beta } ) for faint galaxies at z \simeq 7 , the first meaningful measurements of \langle \beta \rangle at z \simeq 8 , and tentative estimates for a new sample of galaxies at z \simeq 9 . Utilising galaxy selection in the new F140W ( J _ { 140 } ) imaging to minimize colour bias , and applying both colour and power-law estimators of \beta , we find \langle \beta \rangle = -2.1 \pm 0.2 at z \simeq 7 for galaxies with M _ { UV } \simeq - 18 . This means that the faintest galaxies uncovered at this epoch have , on average , UV colours no more extreme than those displayed by the bluest star-forming galaxies at low redshift . At z \simeq 8 we find a similar value , \langle \beta \rangle = -1.9 \pm 0.3 . At z \simeq 9 , we find \langle \beta \rangle = -1.8 \pm 0.6 , essentially unchanged from z \simeq 6 - 7 ( albeit highly uncertain ) . Finally , we show that there is as yet no evidence for a significant intrinsic scatter in \beta within our new , robust z \simeq 7 galaxy sample . Our results are most easily explained by a population of steadily star-forming galaxies with either \simeq solar metallicity and zero dust , or moderately sub-solar ( \simeq 10 - 20 % ) metallicity with modest dust obscuration ( A _ { V } \simeq 0.1 - 0.2 ) . This latter interpretation is consistent with the predictions of a state-of-the-art galaxy-formation simulation , which also suggests that a significant population of very-low metallicity , dust-free galaxies with \beta \simeq - 2.5 may not emerge until M _ { UV } > -16 , a regime likely to remain inaccessible until the James Webb Space Telescope .