  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
$N_{\rm H} \approx4\times10^{21}$ 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 $y_{\rm
ign}\approx1.5\times10^9$ 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.
