Quasar absorption lines provide a precise test of whether the fine-structure constant, α, is the same in different places and through cosmological time. We present a new analysis of a large sample of quasar absorption-line spectra obtained using the Ultraviolet and Visual Echelle Spectrograph (UVES) on the Very Large Telescope (VLT) in Chile. We apply the many-multiplet method to derive values of Δα/α≡ (αz−α0)/α0 from 154 absorbers, and combine these values with 141 values from previous observations at the Keck Observatory in Hawaii. In the VLT sample, we find evidence that α increases with increasing cosmological distance from Earth. However, as previously shown, the Keck sample provided evidence for a smaller α in the distant absorption clouds. Upon combining the samples, an apparent variation of α across the sky emerges which is well represented by an angular dipole model pointing in the direction RA = 17.3 ± 1.0 h and Dec. =−61°± 10°, with amplitude . The dipole model is required at the 4.1σ statistical significance level over a simple monopole model where α is the same across the sky (but possibly different from the current laboratory value). The data sets reveal remarkable consistencies: (i) the directions of dipoles fitted to the VLT and Keck samples separately agree; (ii) the directions of dipoles fitted to z < 1.6 and z > 1.6 cuts of the combined VLT+Keck samples agree; and (iii) in the equatorial region of the dipole, where both the Keck and VLT samples contribute a significant number of absorbers, there is no evidence for inconsistency between Keck and VLT. The amplitude of the dipole is clearly larger at higher redshift. Assuming a dipole-only (i.e. no-monopole) model whose amplitude grows proportionally with ‘lookback-time distance’ (r=ct, where t is the lookback time), the amplitude is (1.1 ± 0.2) × 10−6 GLyr−1 and the model is significant at the 4.2σ confidence level over the null model (Δα/α≡ 0). We apply robustness checks and demonstrate that the dipole effect does not originate from a small subset of the absorbers or spectra. We present an analysis of systematic effects, and are unable to identify any single systematic effect which can emulate the observed variation in α. To the best of our knowledge, this result is not in conflict with any other observational or experimental result.