We have modelled the spectral energy distributions of the 13 Hubble Deep Field (HDF) galaxies reliably detected by the Infrared Space Observatory (ISO). For two galaxies the emission detected by ISO is consistent with being starlight or the infrared `cirrus' in the galaxies. For the remaining 11 galaxies there is a clear mid-infrared excess, which we interpret as emission from dust associated with a strong starburst. 10 of these galaxies are spirals or interacting pairs, while the remaining one is an elliptical with a prominent nucleus and broad emission lines. We give a new discussion of how the star formation rate can be deduced from the far-infrared luminosity, and derive star formation rates for these galaxies of 8-1000phi Msolar yr^-1, where phi takes account of the uncertainty in the initial mass function. The HDF galaxies detected by ISO are clearly forming stars at a prodigious rate compared with nearby normal galaxies. We discuss the implications of our detections for the history of star and heavy element formation in the Universe. Although uncertainties in the calibration, reliability of source detection, associations and starburst models remain, it is clear that dust plays an important role in star formation out to redshift 1 at least.