We have created a high-density gas of interacting positronium (Ps) atoms by irradiating a thin film of nanoporous silica with intense positron bursts and measured the Ps lifetime using a new single-shot technique. When the positrons were compressed to 3.3 x 10(10) cm-2, the apparent intensity of the orthopositronium lifetime component was found to decrease by 33%. We believe this is due to a combination of spin exchange quenching and PS2 molecule formation associated with colliding pairs of oppositely polarized triplet positronium atoms. Our data imply an effective cross section for this process of 2.9 x 10(-14) cm-2.