Many autonomously replicating genetic elements exist as multiple copies within the cell. The copy number of these elements is often assumed to have important fitness consequences for both element and host, yet the forces shaping its evolution are not well understood. The 2µm is a multi-copy plasmid of Saccharomyces yeasts, encoding just four genes that are solely involved in plasmid replication. One simple model for the fitness relationship between yeasts and 2µm is that plasmid copy number evolves as a tradeoff between selection for increased vertical transmission, favoring high copy number, and selection for decreased virulence, favoring low copy number. To test this model, we experimentally manipulated the copy number of the plasmid and directly measured the fitness cost, in terms of growth rate reduction, associated with high plasmid copy number. We find that the fitness burden imposed by the 2µm increases with plasmid copy number, such that each copy imposes a fitness burden of 0.17% (±0.008%), greatly exceeding the cost expected for it to be stably maintained in yeast populations. Our results demonstrate the crucial importance of copy number in the evolution of yeast/2µm associations, and pave the way for future studies examining how selection can shape the cost of multi-copy elements.