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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 two-week 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 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=00^h29^m03.08^s, and Dec=+59°34'19.2" (0.6'' error), compatible with the optical and radio ones. 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 (∼1keV, interpreted as a hot spot on 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 ∼1keV thermal component, while we detected a colder thermal component (∼0.4keV) that we interpret as the emission from the cold disc. A hint of a 6.4keV iron line was detected, together with an excess around 6.8keV and absorption feature around 7.1keV. The last two features have never been detected in the spectra of accretion-driven millisecond pulsars before and, if confirmed, would suggest the presence of an expanding hot corona with high outflow velocities.