Context: Aims: This work presents the results obtained by CoRoT on HD 50844, the only δ Sct star observed in the CoRoT initial run (57.6 d). The aim of these CoRoT observations was to investigate and characterize for the first time the pulsational behaviour of a δ Sct star, when observed at a level of precision and with a much better duty cycle than from the ground. Methods: The 140 016 datapoints were analysed using independent approaches (SigSpec software and different iterative sine-wave fittings) and several checks performed (splitting of the timeseries in different subsets, investigation of the residual light curves and spectra). A level of 10-5 mag was reached in the amplitude spectra of the CoRoT timeseries. The space monitoring was complemented by ground-based high-resolution spectroscopy, which allowed the mode identification of 30 terms. Results: The frequency analysis of the CoRoT timeseries revealed hundreds of terms in the frequency range 0-30 d-1. All the cross-checks confirmed this new result. The initial guess that δ Sct stars have a very rich frequency content is confirmed. The spectroscopic mode identification gives theoretical support since very high-degree modes (up to ℓ=14) are identified. We also prove that cancellation effects are not sufficient in removing the flux variations associated to these modes at the noise level of the CoRoT measurements. The ground-based observations indicate that HD 50844 is an evolved star that is slightly underabundant in heavy elements, located on the Terminal Age Main Sequence. Probably due to this unfavourable evolutionary status, no clear regular distribution is observed in the frequency set. The predominant term (f_1=6.92 d-1) has been identified as the fundamental radial mode combining ground-based photometric and spectroscopic data. Conclusions: The CoRoT space mission was developed and is operated by the French space agency CNES, with participation of ESA's RSSD and Science Programmes, Austria, Belgium, Brazil, Germany, and Spain. This work is based on ground-based observations made with ESO telescopes at the La Silla Observatory under the ESO Large Programme LP178.D-0361 and on data collected at the Observatorio de Sierra Nevada (Spain), at the Observatorio Astronómico Nacional San Pedro Mártir (Mexico), and at the Piszkéstetö Mountain Station of Konkoly Observatory (Hungary). Table 2 is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/506/85 Current address: Laboratoire AIM, CEA/DSM CNRS Université Paris Diderot, CEA, IRFU, SAp, centre de Saclay, 91191 Gif-sur-Yvette, France.