We forecast dark energy constraints that could be obtained from a new large sample of Type Ia supernovae where those at high redshift are acquired with the Euclid space mission. We simulate a three-prong SN survey: a z <0.35 nearby sample (8000 SNe), a 0.2 <z <0.95 intermediate sample (8800 SNe), and a 0.75 <z <1.55 high-z sample (1700 SNe). The nearby and intermediate surveys are assumed to be conducted from the ground, while the high-z is a joint ground-and space-based survey. This latter survey, the "Dark Energy Supernova Infra-Red Experiment" (DESIRE), is designed to fit within 6 months of Euclid observing time, with a dedicated observing programme. We simulate the SN events as they would be observed in rolling-search mode by the various instruments, and derive the quality of expected cosmological constraints. We account for known systematic uncertainties, in particular calibration uncertainties including their contribution through the training of the supernova model used to fit the supernovae light curves. Using conservative assumptions and a 1D geometric Planck prior, we find that the ensemble of surveys would yield competitive constraints: a constant equation of state parameter can be constrained to sigma(omega) = 0.022, and a Dark Energy Task Force figure of merit of 203 is found for a two-parameter equation of state. Our simulations thus indicate that Euclid can bring a significant contribution to a purely geometrical cosmology constraint by extending a high-quality SN Ia Hubble diagram to z similar to 1.5. We also present other science topics enabled by the DESIRE Euclid observations.