We take advantage of the exceptional photometric coverage provided by the combination of GALEX data in the ultraviolet (UV) and the ALHAMBRA survey in the optical and near-infrared to analyse the physical properties of a sample of 1225 GALEX-selected Lyman break galaxies (LBGs) at 0.8 ≲ z ≲ 1.2 that are located in the COSMOS field. This is the largest sample of LBGs studied in this redshift range to date. According to a spectral energy distribution (SED) fitting with synthetic stellar population templates, we find that LBGs at z ˜ 1 are mostly young galaxies with a median age of 341 Myr and have intermediate dust attenuation, ˜ 0.20. Owing to the selection criterion, LBGs at z ˜ 1 are UV-bright galaxies and have a high dust-corrected total star formation rate (SFR), with a median value of 16.9 M⊙ yr-1. Their median stellar mass is log (M*/M⊙) = 9.74. We find that the dust-corrected total SFR of LBGs increases with stellar mass and that the specific SFR is lower for more massive galaxies (downsizing scenario). Only 2 per cent of the galaxies selected through the Lyman break criterion have an active galactic nucleus nature. LBGs at z ˜ 1 are located mostly over the blue cloud of the colour-magnitude diagram of galaxies at their redshift, with only the oldest and/or the dustiest deviating towards the green valley and red sequence. Morphologically, 69 per cent of LBGs are disc-like galaxies, with the fractions of interacting, compact, or irregular systems being much lower, below 12 per cent. LBGs have a median effective radius of 2.5 kpc, and larger galaxies have a higher total SFR and stellar mass. Compared with their high-redshift analogues, we find evidence that LBGs at lower redshifts are larger, redder in the UV continuum, and have a major presence of older stellar populations in their SEDs. However, we do not find significant differences in the distributions of stellar mass or dust attenuation.