We use the new ultra-deep, near-infrared imaging of the Hubble Ultra-Deep Field (HUDF) provided by our UDF12 HST WFC3/IR campaign to explore the rest-frame UV properties of galaxies at redshifts z > 6.5. We present the first unbiased measurement of the average UV power-law index, beta, for faint galaxies at z ~ 7, the first meaningful measurements of beta at z ~ 8, and tentative estimates for a new sample of galaxies at z ~ 9. Utilising galaxy selection in the new F140W imaging to minimize colour bias, and applying both colour and power-law estimators of beta, we find beta = -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, beta = -1.9 (+/-0.3). At z ~ 9, we find beta = -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 beta 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 beta ~ -2.5 may not emerge until M_UV > -16, a regime likely to remain inaccessible until the James Webb Space Telescope.
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