We have detected seven different molecular lines in absorption towards the remote radio source 1504+377. The absorption occurs in an edge-on galaxy at a redshift z=0.672. The absorption splits into several components, gathered in two groups separated by 330km/s. The two groups are interpreted as coming from a spiral arm and a nuclear ring in non-circular motion. Given the spatial coincidence of the remote radio source, and the galaxy as well as the absence of gravitationally lensed images, the absorption is likely to originate in the galaxy hosting the AGN. Comparing different lines, we find a large density for the absorbing molecular clouds, and an excitation temperature of 10-20K. The HCO+ abundance is enhanced by at least one order of magnitude with respect to current chemical equilibrium models. The HNC/HCN ratio yields a kinetic temperature in accordance with the excitation temperature. This suggests a dense medium, however with a low column density, since the absorption lines are optically thin. A molecular cloud composed of small dense clumps with a low filling factor could account for that. We also present a list of upper limits for additional candidates searched for molecular absorption. With previously published negative results, the total number of objects searched is about 50, while 4 absorption systems have been found. We discuss the efficiency of various selection criteria.