Context . The small- to intermediate-mass (M <0.8 M ⊙ ), most metal-poor stars that formed in the infancy of the Universe are still shining today in the sky. They are very rare, but their discovery and investigation brings new knowledge on the formation of the first stellar generations. Aims . SDSS J102915.14+172927.9 is one of the most metal-poor star known to date. Since no carbon can be detected in its spectrum, a careful upper limit is important, both to classify this star and to distinguish it from the carbon-enhanced stars that represent the majority at these metallicities. Methods . We undertook a new observational campaign to acquire high-resolution UVES spectra. The new spectra were combined with archival spectra in order to increase the signal-to-noise ratio. From the combined spectrum, we derived abundances for seven elements (Mg, Si, Ca, Ti, Fe, Ni, and a tentative Li) and five significant upper limits (C, Na, Al, Sr, and Ba). Results . The star has a carbon abundance A(C) <4.68 and therefore is not enhanced in carbon, at variance with the majority of the stars at this Fe regime, which typically show A(C)> 6.0. A feature compatible with the Li doublet at 670.7 nm is tentatively detected. Conclusions . The upper limit on carbon implies Z<1.915 × 10 −6 , more than 20 times lower than the most iron-poor star known. Therefore, the gas cloud out of which the star was formed did not cool via atomic lines but probably through dust. Fragmentation of the primordial cloud is another possibility for the formation of a star with a metallicity this low.