Brown dwarfs and giant planets orbiting close to a host star are subjected to significant irradiation that can modify the properties of their atmospheres. In order to test the atmospheric models that are used to describe these systems, it is necessary to obtain accurate observational estimates of their physical properties (masses, radii, temperatures, albedos). Interacting compact binary systems provide a natural laboratory for studying strongly irradiated sub-stellar objects. As the mass-losing secondary in these systems makes a critical, but poorly understood transition from the stellar to the sub-stellar regime, it is also strongly irradiated by the compact accretor. In fact, the internal and external energy fluxes are both expected to be comparable in these objects, providing access to an unexplored irradiation regime. However, the atmospheric properties of such donors have so far remained largely unknown 3 . Here, we report the direct spectroscopic detection and characterisation of an irradiated sub-stellar donor in an accreting white dwarf binary system. Our near-infrared observations allow us to determine a model-independent mass estimate for the donor of M 2 = 0.055 ± 0.008M and an average spectral type of L1 ± 1, supporting both theoretical predictions and model-dependent observational constraints. Our time-resolved data also allow us to estimate the average irradiation-induced temperature difference between the day and night sides on the sub-stellar donor, ∆T 57 K, and the maximum difference between the hottest and coolest parts of its surface, of ∆T max 200 K. The observations are well described by a simple geometric reprocessing model with a bolometric (Bond) albedo of A B < 0.54 at the 2-σ confidence level, consistent with high reprocessing efficiency, but poor lateral heat redistribution in the donor's atmosphere 4, 5 .Only a single brown dwarf donor to an accreting white dwarf has been detected spectroscopically to date 3 , and no empirical information at all is available regarding the effect of irradiation on the donor's atmosphere. We have therefore carried out simultaneous optical and nearinfrared time-resolved spectroscopy of one such system with the X-Shooter instrument 6 at the Very 2 Large Telescope (ESO). Our target, SDSS J143317.78+101123.3 7 (J1433 hereafter), is an eclipsing system with a short orbital period (P orb = 78.1 min) and a likely donor mass well below the hydrogen-burning limit, as estimated from eclipse modelling (M 2 = 0.0571 ± 0.0007M 8, 9 ). The exceptionally wide wavelength coverage provided by our data set (0.35 µm -2.5 µm) allows us to confidently dissect the overall spectral energy distribution. This is illustrated in Fig. 1. Diskformed, double-peaked hydrogen emission lines can be seen across the entire spectral range. The presence of broad Balmer absorption lines between 0.3 µm and 0.5 µm shows that the white dwarf is the strongest emitter in this wave band. However, the donor star spectrum is clearly visible in the near-infrared and dominates between 1.0 µm and 2.5...