We take advantage of the exceptional photometric coverage provided by the combination of GALEX data in the UV and the ALHAMBRA survey in the optical and near-IR to analyze the physical properties of a sample of 1225 GALEX-selected Lyman break galaxies (LBGs) at $0.8 \lesssim z \lesssim 1.2$ located in the COSMOS field. This is the largest sample of LBGs studied at that redshift range so far. 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, $\ ∼ 0.20$. Due to their selection criterion, LBGs at $z ∼ 1$ are UV-bright galaxies and have high dust-corrected total SFR, with a median value of 16.9 $M_⊙ {\rm yr}^{-1}$. Their median stellar mass is $\log{\left(M_*/M_\odot \right)} = 9.74$. We obtain that the dust-corrected total SFR of LBGs increases with stellar mass and the specific SFR is lower for more massive galaxies. Only 2% of the galaxies selected through the Lyman break criterion have an AGN nature. LBGs at $z ∼ 1$ are mostly located over the blue cloud of the color-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% of LBGs are disk-like galaxies, with the fraction of interacting, compact, or irregular systems being much lower, below 12%. LBGs have a median effective radius of 2.5 kpc and bigger galaxies have higher total SFR and stellar mass. Comparing to their high-redshift analogues, we find evidence that LBGs at lower redshifts are bigger, redder in the UV continuum, and have a major presence of older stellar populations in their SEDs. However, we do not find significant difference in the distributions of stellar mass or dust attenuation. ; Comment: Accepted for publication in MNRAS