NGC346-013 is a peculiar double-lined eclipsing binary in the Small Magellanic Cloud discovered by the VLT-FLAMES survey of massive stars. Spectra obtained with VLT-FLAMES are used to construct a radial velocity curve and photometry obtained with the Faulkes Telescope South is then used to derive orbital parameters, while spectra of the secondary are compared with synthetic spectra from TLUSTY model atmospheres. The orbital period is found to be 4.20381(12) days, with masses of 19.1+/-1.0 and

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... .9+/-0.6 Msun. The primary is a rapidly rotating late-O dwarf while the secondary, an early-B giant, displays near-synchronous rotation and has filled its Roche lobe, implying that it was originally the more massive component with recent mass transfer 'spinning up' the primary to near-critical rotation. Comparison with synthetic spectra finds temperatures of 34.5kK and 24.5kK for the primary and secondary respectively, with the nitrogen abundance of the secondary enhanced compared to baseline values for the SMC, consistent with the predictions of models of interacting binaries. NGC346-013 likely evolved via non-conservative mass transfer in a system with initial masses ~22+15Msun, with the well-constrained orbital solution and atmospheric parameters making it an excellent candidate for tailored modelling with binary evolution codes. This system will form a cornerstone in constraining the physics of thermal timescale mass transfer, and the associated mass transfer efficiency, in massive close binary systems.