A search has been carried out for the 234 GHz N = 2-0, J = 1-1 transition of (O-16)(O-18) using the 13.7 m FCRAO radio telescope. No emission was detected toward six giant molecular clouds. Observations of the 220 GHz J = 2-1 transition of (C-18)O yield column densities for this species 1-3 x 10 to the 16th per sq cm; the resulting limits on the (O2)/(CO) ratio lie between less than 0.5 and less than 4. According to various chemical models, the ratio of molecular oxygen to carbon monoxide is primarily sensitive to the age of a cloud and to its carbon to oxygen ratio. For ages exceeding 3 x 10 to the 16th yr and total carbon-to-oxygen ratio of less than (O2)/(CO) can approach unity. The limits can be interpreted as indicating that the observed clouds are not chemically 'mature' or that (carbon)/(oxygen) is greater than 1. However, significant exploitation of molecular oxygen as a tracer of cloud structure and evolution will require more sensitive observations, which may be best carried out from above earth's atmosphere.