The high-frequency-peaked BL Lac (HBL) 1ES 0806+524 (z = 0.138) was discovered in VHE $gamma$ rays in 2008. Until now, the broad-band spectrum of 1ES 0806+524 has been only poorly characterized, in particular at high energies. We analysed multiwavelength observations from $gamma$ rays to radio performed from 2011 January to March, which were triggered by the high activity detected at optical frequencies. These observations constitute the most precise determination of the broad-band emission of 1ES 0806+524 to date. The stereoscopic MAGIC observations yielded a $gamma$-ray signal above 250 GeV of $(3.7 pm 0.7)$ per cent of the Crab Nebula flux with a statistical significance of 9.9 $sigma$. The multiwavelength observations showed significant variability in essentially all energy bands, including a VHE $gamma$-ray flare that lasted less than one night, which provided unprecedented evidence for short-term variability in 1ES 0806+524. The spectrum of this flare is well described by a power law with a photon index of $2.97 pm 0.29$ between $sim$150 GeV and 1 TeV and an integral flux of $(9.3 pm 1.9)$ per cent of the Crab Nebula flux above 250 GeV. The spectrum during the non-flaring VHE activity is compatible with the only available VHE observation performed in 2008 with VERITAS when the source was in a low optical state. The broad-band spectral energy distribution can be described with a one-zone Synchrotron Self Compton model with parameters typical for HBLs, indicating that 1ES 0806+524 is not substantially different from the HBLs previously detected.
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