We report on an observation of the recently discovered accreting millisecond X-ray pulsar IGR J00291+5934 performed with the RXTE-Proportional Counter Array (PCA) and Chandra-High Energy Transmission Grating Spectrometer (HETGS). The RXTE data are from a twoweek follow-up of the source, while the Chandra observation took place around the end of the follow-up, about 12 days after the discovery of\ud the source, when the source flux had decreased already by a factor of ten. The analysis of the Chandra data allowed us to extract the most precise X-ray position of IGR J00291+5934, RA = 00h 29m 03.08s, and Dec =+59◦ 34 19.2 (0.6 error), compatible with the optical and radio ones.\ud We find that the spectra of IGR J00291+5934 can be described by a combination of a thermal component and a power-law. Along the outburst detected by PCA, the power-law photon index showed no particular trend, while the thermal component (∼1 keV, interpreted as a hot spot on\ud the neutron star surface) became weaker until non-detection. In the simultaneous observation of the weak Chandra /RXTE spectrum, there was no longer any indication of the ∼1 keV thermal component, while we detected a colder thermal component (∼0.4 keV) that we interpret as the\ud emission from the cold disc. A hint of a 6.4 keV iron line was detected, together with an excess around 6.8 keV and absorption feature around 7.1 keV. The last two features have never been detected in the spectra of accretion-driven millisecond pulsars before and, if confirmed, would\ud suggest the presence of an expanding hot corona with high outflow velocities.\u
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