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, β , (f λ ∝ λ β ) for faint galaxies at z ≃ 7, the first meaningful measurements of β at z ≃ 8, and tentative estimates for a new sample of galaxies at z ≃ 9. Utilising galaxy selection in the new F140W (J 140 ) imaging to minimize colour bias, and applying both colour and power-law estimators of β, we find β = −2.1 ± 0.2 at z ≃ 7 for galaxies with M UV ≃ −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 ≃ 8 we find a similar value, β = −1.9 ± 0.3. At z ≃ 9, we find β = −1.8 ± 0.6, essentially unchanged from z ≃ 6 − 7 (albeit highly uncertain). Finally, we show that there is as yet no evidence for a significant intrinsic scatter in β within our new, robust z ≃ 7 galaxy sample. Our results are most easily explained by a population of steadily star-forming galaxies with either ≃ solar metallicity and zero dust, or moderately sub-solar (≃ 10 − 20%) metallicity with modest dust obscuration (A V ≃ 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 β ≃ −2.5 may not emerge until M UV > −16, a regime likely to remain inaccessible until the James Webb Space Telescope.