We have carried out high angular resolution near-infrared imaging and low-resolution (R˜1000) spectroscopy of the nearest known brown dwarf, varepsilon Indi B, using the ESO VLT NAOS/CONICA adaptive optics system. We find it to be a close binary (as also noted by Volk et al. 2003), with an angular separation of 0.732 arcsec, corresponding to 2.65 AU at the 3.626 pc distance of the varrepsilon Indi system. In our discovery paper (Scholz et al. 2003), we concluded that varepsilon Indi B was a ˜50 MJup T2.5 dwarf: our revised finding is that the two system components (varepsilon Indi Ba and varrepsilon Indi Bb) have spectral types of T1 and T6, respectively, and estimated masses of 47 and 28 MJup, respectively, assuming an age of 1.3 Gyr. Errors in the masses are ±10 and ±7 MJup, respectively, dominated by the uncertainty in the age determination (0.8-2 Gyr range). This uniquely well-characterised T dwarf binary system should prove important in the study of low-mass, cool brown dwarfs. The two components are bright and relatively well-resolved: varepsilon Indi B is the only T dwarf binary in which spectra have been obtained for both components. The system has a well-established distance and age. Finally, their orbital motion can be measured on a fairly short timescale (nominal orbital period ˜15 yrs), permitting an accurate determination of the true total system mass, helping to calibrate brown dwarf evolutionary models. Based on observations collected with the ESO VLT, Paranal, Chile.