The abundance of molecular oxygen in dense molecular clouds is a key problem within astrochemistry. The interstellar medium is thought to be oxygen-rich (elemental C/O=0.5) and all theoretical models predict that the oxygen not locked up in CO will be in O2. However, despite sensitive searches, gaseous O2 has never been detected in the interstellar medium or in other galaxies. The derived upper limits are in the order of O2/CO < 0.01. Hence it has been suggested that the excess O is frozen out in icy grain mantles. Theoretical models of icy grain mantles predict that O2 can be an important grain mantle molecule in apolar ices in dense molecular clouds. Recent laboratory data show that O2 can be detected directly through the weak fundamental transition at 6.4 mu or by observing the photolysis product O3 at 9.7 mu. Large spectroscopic databases on CO and CO2 ices as well as isolated water ice - created for ISO - revealed that O2 leaves its "fingerprints" in the profiles of the strong CO and CO2 bands and shifts strongly the position of isolated water ice bands. This combined spectroscopic information can be used as a powerful tool to make the search for solid molecular oxygen with ISO successful. We propose high resolution spectroscopic observations with the ISO satellite towards a number of embedded sources and star-forming regions in order to detect solid molecular oxygen and the sharp transition of its photolysis product ozone. For our objects which have a typical visual extinction of 20 mag, we would be able to detect the solid O2 feature if the solid O2 abundance is larger than 10 % of the elemental O. The detection of solid molecular oxygen and ozone and the estimation of their abundances provide strong constraints for the oxygen budget in the interstellar medium. New laboratory data on solid O2 are the basis of this extended version related to the previous proposal OXYOZONE/Open time, accepted in Cat.A.