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. During the second commissioning run of the GIANO high-resolution spectrograph at the 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 and includes the whole H-band. 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). The other lines are attributable to O_2_ or unidentified. Several of the faint lines are in spectral regions that were previously believed to be free of lines emission. The continuum in the H-band is marginally detected at a level of about 300photons/m^2^/s/arcsec^2^/um. The observed spectrum and the list of observed sky-lines are published in electronic format. 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).