The rupture of fluid necks in a surrounding liquid in the presence of surface-active polymers shows a variety of changes with respect to the pure-fluid case. The role of the surface rheology due solely to the presence of such agents at the surface is uncovered by studying how the fluid neck radius approaches zero at pinch-off. The thinning dynamics of the neck turn out to be exponential in time. The thinning rate extracted from this dynamics is related to the surface viscosity of the polymer layer coating the interface and estimates of this quantity are presented.