The interconversion of the two allotropes of the hydrogen molecule (para-H2 and ortho-H2) incarcerated inside the fullerene C60 is reported (oH2@C60 and pH2@C60, respectively). For conversion, oH2@C60 was adsorbed at the external surface of the zeolite NaY and immersed into liquid oxygen at 77 K. Equilibrium was reached in less than 0.5 h. Rapid removal of oxygen provides a sample of enriched pH2@C60 that is stable for many days in the absence of paramagnetic catalysts (half-life approximately 15 days). Enriched pH2@C60 is nonvolatile and soluble in organic solvents. At room temperature in the presence of a paramagnetic catalyst (dissolved O2 or the nitroxide Tempo) a slow back conversion into oH2@C60 was observed by 1H NMR. A bimolecular rate constant for conversion of pH2@C60 to oH2@C60 using Tempo of kTempo approximately 4 x 10-5 M-1 s-1 was observed, which is approximately 3 orders of magnitudes slower than that for dissolved pH2 in organic solvents which is not protected by the C60 shell.