The bacterial ubiD and ubiX or the homologous fungal fdc1 and pad1 genes have been implicated in the non-oxidative reversible decarboxylation of aromatic substrates, and play a pivotal role in bacterial ubiquinone (also known as coenzyme Q) biosynthesis(1-3) or microbial biodegradation of aromatic compounds(4-6), respectively. Despite biochemical studies on individual gene products, the composition and cofactor requirement of the enzyme responsible for in vivo decarboxylase activity remained unclear(7-9). Here we show that Fdc1 is solely responsible for the reversible decarboxylase activity, and that it requires a new type of cofactor: a prenylated flavin synthesized by the associated UbiX/Pad1(10). Atomic resolution crystal structures reveal that two distinct isomers of the oxidized cofactor can be observed, an isoalloxazine N5-iminium adduct and a N5 secondary ketimine species with markedly altered ring structure, both having azomethine ylide character. Substrate binding positions the dipolarophile enoic acid group directly above the azomethine ylide group. The structure of a covalent inhibitor-cofactor adduct suggests that 1,3-dipolar cycloaddition chemistry supports reversible decarboxylation in these enzymes. Although 1,3-dipolar cycloaddition is commonly used in organic chemistry(11,12), we propose that this presents the first example, to our knowledge, of an enzymatic 1,3-dipolar cycloaddition reaction. Our model for Fdc1/UbiD catalysis offers new routes in alkene hydrocarbon production or aryl (de) carboxylation.