In many viruses, molecular motors forcibly pack single DNA molecules to near-crystalline density into 1/450-100 nm prohead shells. Unexpectedly, we found that packaging frequently stalls in conditions that induce net attractive DNA-DNA interactions. Here, we present findings suggesting that this stalling occurs because the DNA undergoes a nonequilibrium jamming transition analogous to that observed in many soft-matter systems, such as colloidal and granular systems. Experiments in which conditions are changed during packaging to switch DNA-DNA interactions between purely repulsive and net attractive reveal strongly history-dependent dynamics. An abrupt deceleration is usually observed before stalling, indicating that a transition in DNA conformation causes an abrupt increase in resistance. Our findings suggest that the concept of jamming can be extended to a single polymer molecule. However, compared with macroscopic samples of colloidal particles we find that single DNA molecules jam over a much larger range of densities. We attribute this difference to the nanoscale system size, consistent with theoretical predictions for jamming of attractive athermal particles.