We measure the fraction of galaxies undergoing disk-disk major mergers (f mph m ) at intermediate redshifts (0.35 <= z < 0.85) by studying the asymmetry index A of galaxy images. Results are provided for B- and K s-band absolute magnitude selected samples from the Groth strip in the galaxy origins and young assembly photometric survey. Three sources of systematic error are carefully addressed and quantified. The effects of the large errors in the photometric redshifts and asymmetry indices are corrected with maximum-likelihood techniques. Biases linked to the redshift degradation of the morphological information in the images are treated by measuring asymmetries on images artificially redshifted to a reference redshift of zd = 0.75. Morphological K-corrections are further constrained by remaining within redshifts where the images sample redward of 4000 Å. We find that: (1) our data allow for a robust merger fraction to be provided for a single redshift bin, which we center at z = 0.6. (2) Merger fractions at that z have lower values than previous determinations: f mph m = 0.045+0.014 -0.011 for MB <= -20 galaxies, and f mph m = 0.031+0.013 -0.009 for M_{K_s} ≤ -23.5 galaxies. And, (3) failure to address the effects of the large observational errors leads to overestimating f mph m by factors of 10%-60%. Combining our results with those on other B-band selected samples, and parameterizing the merger fraction evolution as f mph m (z) = f mph m (0)(1 + z) m , we obtain that m = 2.9 ± 0.8, and f mph m (0) = 0.012 ± 0.004. For an assumed merger timescale between 0.35-0.6 Gyr, these values imply that only 20%-35% of present-day MB <= -20 galaxies have undergone a disk-disk major merger since z ~ 1. Assuming a K s-band mass-to-light ratio not varying with luminosity, we infer that the merger rate of galaxies with stellar mass M sstarf gsim 3.5 × 1010 M sun is real m = 1.6+0.9 -0.6 × 10-4 Mpc-3 Gyr-1 at z = 0.6. When we compare with previous studies at similar redshifts, we find that the merger rate decreases when mass increases.