Context. The extragalactic very-high-energy gamma-ray sky is rich in blazars. These are jetted active galactic nuclei that are viewed at a small angle to the line-of-sight. Only a handful of objects viewed at a larger angle are so far known to emit above 100 GeV. Multi-wavelength studies of such objects up to the highest energies provide new insights into the particle and radiation processes of active galactic nuclei. Aims: We aim to report the results from the first multi-wavelength campaign observing the TeV detected nucleus of the active galaxy IC 310, whose jet is observed at a moderate viewing angle of 10°-20°. Methods: The multi-instrument campaign was conducted between 2012 November and 2013 January, and involved observations with MAGIC, Fermi, INTEGRAL, Swift, OVRO, MOJAVE and EVN. These observations were complemented with archival data from the AllWISE and 2MASS catalogs. A one-zone synchrotron self-Compton model was applied to describe the broadband spectral energy distribution. Results: IC 310 showed an extraordinary TeV flare at the beginning of the campaign, followed by a low, but still detectable TeV flux. Compared to previous measurements in this energy range, the spectral shape was found to be steeper during the low emission state. Simultaneous observations in the soft X-ray band showed an enhanced energy flux state and a harder-when-brighter spectral shape behavior. No strong correlated flux variability was found in other frequency regimes. The broadband spectral energy distribution obtained from these observations supports the hypothesis of a double-hump structure. Conclusions: The harder-when-brighter trend in the X-ray and VHE emission, observed for the first time during this campaign, is consistent with the behavior expected from a synchrotron self-Compton scenario. The contemporaneous broadband spectral energy distribution is well described with a one-zone synchrotron self-Compton model using parameters that are comparable to those found for other gamma-ray-emitting misaligned blazars.