Context. The early-type binary MY Cam belongs to the young open cluster Alicante 1, embedded in Cam OB3. Aims: MY Cam consists of two early-O type main-sequence stars and shows a photometric modulation suggesting an orbital period slightly above one day. We intend to confirm this orbital period and derive orbital and stellar parameters. Methods: Timing analysis of a very exhaustive (4607 points) light curve indicates a period of 1.1754514 ± 0.0000015 d. High-resolution spectra and the cross-correlation technique implemented in the todcor program were used to derive radial velocities and obtain the corresponding radial velocity curves for MY Cam. Modelling with the stellar atmosphere code fastwind was used to obtain stellar parameters and create templates for cross-correlation. Stellar and orbital parameters were derived using the Wilson-Devinney code, such that a complete solution to the binary system could be described. Results: The determined masses of the primary and secondary stars in MY Cam are 37.7 ± 1.6 and 31.6 ± 1.4M⊙, respectively. The corresponding temperatures, derived from the model atmosphere fit, are 42 000 and 39 000 K, with the more massive component being hotter. Both stars are overfilling their Roche lobes, sharing a common envelope. Conclusions: MY Cam contains the most massive dwarf O-type stars found so far in an eclipsing binary. Both components are still on the main sequence, and probably not far from the zero-age main sequence. The system is a likely merger progenitor, owing to its very short period. Tables 1 and 3 are available in electronic form at http://www.aanda.orgPhotometric data (Table 2) are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/572/A110