When a droplet of water impacts a hydrophobic surface, the drop is often observed to bounce. However, for about 10 years it has been known that the addition of very small quantities (approximately 100 ppm) of a flexible polymer such as poly-(ethylene oxide) can completely prevent rebound. This effect has for some time been explained in terms of the stretching of polymer chains by a velocity gradient in the fluid, resulting in a transient increase in the so-called "extensional viscosity." Here we show, by measuring the fluid velocity inside the impacting drop, that the extensional viscosity plays no role in the antirebound phenomenon. Using fluorescently labeled lambda DNA we demonstrate that the observed effect is due to the stretching of polymer molecules as the droplet edge sweeps the substrate, retarding the movement of the receding contact line.