Virtis-H is the high spectral resolution channel of the visible and infrared imaging spectrometer VIRTIS, an instrument of the ESA/ROSETTA mission devoted to the in-orbit remote sensing study of the comet P/46 Wirtanen. After successful tests and calibration, the flight model has been delivered to the European Space Agency for integration on the satellite before the launch foreseen in January 2003. The Virtis-H channel is a cross-dispersion spectrometer in the spectral range 2-5um with a resolution between 1200 and 3000. Its design consists in an afocal telescope-collimator off-axis parabola mirrors, a prism-grating system performing the cross-dispersion, and a three-lens objective imaging the entrance slit on a 436x270 HgCdTe array from Raytheon/IRCOE. At each recorded image, a full spectrum of the observed scene is reconstructed allowing the study of the fine spectral details of the coma and the cometary nucleus. The calibration have shown the fully compliance of the instrument performances with the simulations in terms of spectral resolution, radiometric accuracy and sensibility. For example, spectra of gas, water ice and mineral powders have been measured with Virtis-H showing either its ability to resolve fine spectral lines but also its sensitivity to low fluxes; furthermore, measurements on a 250K blackbody shows its sensibility to relative temperature variation lower than 0.5oC..