The R1 plasmid employs ATP-driven polymerisation of the actin-like protein ParM to move newly replicated DNA to opposite poles of a bacterial cell. This process is essential for ensuring accurate segregation of the low-copy number plasmid and is the best characterised example of DNA partitioning in prokaryotes. In vivo, ParM only forms long filaments when capped at both ends by attachment to a centromere-like region parC, through a small DNA-binding protein ParR. Here, we present biochemical and electron microscopy data leading to a model for the mechanism by which ParR-parC complexes bind and stabilise elongating ParM filaments. We propose that the open ring formed by oligomeric ParR dimers with parC DNA wrapped around acts as a rigid clamp, which holds the end of elongating ParM filaments while allowing entry of new ATP-bound monomers. We propose a processive mechanism by which cycles of ATP hydrolysis in polymerising ParM drives movement of ParR-bound parC DNA. Importantly, our model predicts that each pair of plasmids will be driven apart in the cell by just a single double helical ParM filament.