Context. The microquasar GX 339-4 experienced a new outburst in 2010: it was observed simultaneously at various wavelengths from radio up to soft γ-rays. We focus on observations that are quasi-simultaneous with those made with the INTEGRAL and RXTE satellites: these were collected in 2010 March-April during our INTEGRAL target of opportunity programme, and during some of the other INTEGRAL observing programmes with GX 339-4 in the field-of-view. Aims: X-ray transients are extreme systems that often harbour a black hole, and are known to emit throughout the whole electromagnetic spectrum when in outburst. The goals of our programme are to understand the evolution of the physical processes close to the black hole and to study the connections between the accretion and ejection. Methods: We analysed radio, NIR, optical, UV, X-ray and soft γ-ray observations. We studied the source evolution in detail by producing light curves, hardness-intensity diagrams and spectra. We fitted the broadband data with phenomenological, then physical, models to study the emission coming from the distinct components. Results: Based on the energy spectra, the source evolved from the canonical hard state to the canonical soft state. The source showed X-ray spectral variations that were correlated with changes in radio, NIR and optical emission. The bolometric flux increased from 0.8 to 2.9 × 10-8 erg cm-2 s-1 while the relative flux and contribution of the hot medium decreased on the average. Reprocessing in the disc was likely to be strong at the end of our observations. Conclusions: The source showed a behaviour similar to that of previous outbursts, with some small deviations in the hard X-ray parameter evolution. The radio, NIR and optical emission from jets was detected and observed to fade as the source softened. The results are discussed within the context of disc and jet models.