IGR J00291+5934 is the fastest-known accretion-powered X-ray pulsar, discovered during a transient outburst in 2004. In this paper, we report on Integral and Swift observations during the 2015 outburst, which lasts for $sim25$ d. The source has not been observed in outburst since 2008, suggesting that the long-term accretion rate has decreased by a factor of two since discovery. The averaged broad-band (0.1 - 250 keV) persistent spectrum in 2015 is well described by a thermal Comptonization model with a column density of $N_{rm H} approx4times10^{21}$ cm$^{-2}$, a plasma temperature of $kT_{rm e} approx50$ keV, and a Thomson optical depth of $tau_{rm T}approx1$. Pulsations at the known spin period of the source are detected in the Integral data up to the $sim150$ keV energy band. We also report on the discovery of the first thermonuclear burst observed from IGR J00291+5934, which lasts around 7 min and occurs at a persistent emission level corresponding to roughly $1.6%$ of the Eddington accretion rate. The properties of the burst suggest it is powered primarily by helium ignited at a depth of $y_{rm ign}approx1.5times10^9$ g cm$^{-2}$ following the exhaustion by steady burning of the accreted hydrogen. The Swift/BAT data from the first $sim20$ s of the burst provide indications of a photospheric radius expansion phase. Assuming this is the case, we infer a source distance of $d = 4.2 pm 0.3$ kpc.
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