We present deep I-band imaging (NOT-ALFOSC) and spectroscopic (GTC-OSIRIS) observations of the gravitational lens system SDSS J1339+1310. This consists of two images of a lensed quasar, A and B, and a lensing galaxy G between the two quasar images. Our observations led to the following main results: (1) We obtained new accurate positions for B and G (relative to A), as well as structure parameters for the light distribution of G. The new position angle for G is separated by ~50° from the previously determined value. (2) The spectrum of G is typical for early-type galaxies, and we measured its redshift (0.609 ± 0.001) for the first time. (3) We determined the flux ratio B/A for the cores of the emission lines in the two quasar spectra. They were used to constrain the macrolens flux ratio MBA and dust extinction parameters. (4) The continuum flux ratio was appropriately corrected to obtain the microlensing magnification ratio of the continuum μBA. This μBA indicates that B is amplified (relative to A) by a factor of about 3-5, with larger amplifications at shorter wavelengths. The observed microlensing chromaticity coincides with a sharp drop in the r-band flux of B. (5) We reconstructed the lensing mass from the new observational constraints on the relative astrometry, MBA and the luminous structure of G. We also used the redshift of G to predict the time delay between quasar images (43 ± 5 d, A is leading).