The recently demonstrated methods to cool and manipulate neutral molecules offer new possibilities for precision tests of fundamental physics theories. We here discuss the possibility of testing the time-invariance of fundamental constants using near degeneracies between rotational levels in the fine structure ladders of molecular radicals. We show that such a degeneracy occurs between the J = 6, Omega = 1 and the J = 8, Omega = 0 levels of the various natural isotopomers of carbon monoxide in its a(3)Pi state. As a result, the 2-photon transition that connects these states is 300 times more sensitive to a variation of m(p)/m(e) than a pure rotational transition. We present a molecular beam apparatus that might be used to measure these transitions with a fractional accuracy of 10(-12). Ultimately, the precision of an experiment on metastable CO will be limited by the lifetime of the a(3)Pi state. We will discuss other possible molecules that have a suitable level structure and can be cooled using one of the existing methods.
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