Aims Determining the intensity of lines and continuum airglow emission in the H-band is important for the design of faint-object infrared spectrographs. Existing spectra at low/medium resolution cannot disentangle the true sky-continuum from instrumental effects (e.g. diffuse light in the wings of strong lines). We aim to obtain, for the first time, a high resolution infrared spectrum deep enough to set significant constraints on the continuum emission between the lines in the H-band. Methods During the second commissioning run of the GIANO high-resolution infrared spectrograph at La Palma Observatory, we pointed the instrument directly to the sky and obtained a deep spectrum that extends from 0.97 to 2.4 micron. Results The spectrum shows about 1500 emission lines, a factor of two more than in previous works. Of these, 80% are identified as OH transitions; half of these are from highly excited molecules (hot-OH component) that are not included in the OH airglow emission models normally used for astronomical applications. The other lines are attributable to O2 or unidentified. Several of the faint lines are in spectral regions that were previously believed to be free of line emission. The continuum in the H-band is marginally detected at a level of about 300 photons/m^2/s/arcsec^2/micron, equivalent to 20.1 AB-mag/arcsec^2. The observed spectrum and the list of observed sky-lines are published in electronic format. Conclusions Our measurements indicate that the sky continuum in the H-band could be even darker than previously believed. However, the myriad of airglow emission lines severely limits the spectral ranges where very low background can be effectively achieved with low/medium resolution spectrographs. We identify a few spectral bands that could still remain quite dark at the resolving power foreseen for VLT-MOONS (R ~6,600).
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