Many species exhibit high-amplitude, extinction-prone cycles, for instance due to resource enrichment (called the paradox of enrichment). How do such species manage to persist? One possibility is metapopulation dynamics, but it is unclear if these can mitigate the paradox of enrichment. The paradox of enrichment might increase local population extinction rates, and might also increase the spatial synchrony of population fluctuations because population cycles are easily synchronized by even low rates of dispersal. Spatially-synchronous population fluctuations would leave no scope for rescue effects, and spatially-synchronous local extinctions would leave no source for recolonization. We conducted a protist microcosm experiment to test how dispersal and enrichment affect the persistence of predator-prey metapopulations by altering spatial synchrony. We assembled 54, 4-patch metapopulations of the protist predator Euplotes patella and its protist prey Tetrahymena pyriformis. Each metapopulation experienced one of nine dispersal rates, crossed with one of 6 enrichment levels. We tested the effects of enrichment and dispersal on predator-prey cycle amplitude, predator population extinction risk, spatial synchrony of predator population fluctuations, and predator metapopulation persistence time. Enrichment increased predator-prey cycle amplitude but had no detectable effect on population extinction risk. Dispersal had no effect on predator-prey cycle amplitude. Spatial synchrony increased with enrichment and dispersal. Metapopulation persistence decreased with enrichment and spatial synchrony. We conclude that increasing enrichment reduced metapopulation persistence primarily by increasing spatial synchrony of local extinctions, rather than by increasing their frequency. These results highlight the importance of synchrony-preventing mechanisms for metapopulation persistence.