We investigate the conformational changes of a single self-avoiding flexible polymer in strong sedimentation fields using hydrodynamic simulations and scaling arguments. For small fields or short chains hydrodynamic recirculation leads to compaction. For elevated fields or long chains the hydrodynamic drag leads to stretching of the coil and a decrease of the sedimentation coefficient for both linear and circular DNA, in agreement with results in ultracentrifuge experiments. At very large fields a spontaneous symmetry breaking takes place, and the polymer adopts a conformation consisting of a compact head and a trailing straight tail.