We have carried out a spectroscopic variability survey of some of the most massive stars in the Arches cluster, using K-band observations obtained with SINFONI on the VLT. One target, F2, exhibits substantial changes in radial velocity (RV); in combination with new KMOS and archival SINFONI spectra, its primary component is found to undergo RV variation with a period of 10.483 ± 0.002 d and an amplitude of 350 km s-1. A secondary RV curve is also marginally detectable. We reanalysed archival NAOS-CONICA photometric survey data in combination with our RV results to confirm this object as an eclipsing SB2 system, and the first binary identified in the Arches. We have modelled it as consisting of an 82 ± 12 M⊙ WN8-9h primary and a 60 ± 8 M⊙ O5-6 Ia+ secondary, and as having a slightly eccentric orbit, implying an evolutionary stage prior to strong binary interaction. As one of four X-ray bright Arches sources previously proposed as colliding-wind massive binaries, it may be only the first of several binaries to be discovered in this cluster, presenting potential challenges to recent models for the Arches' age and composition. It also appears to be one of the most massive binaries detected to date; the primary's calculated initial mass of ≳120 M⊙ would arguably make this the most massive binary known in the Galaxy. The individual reduced spectra and the two disentangled spectra 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/617/A66.