The FENE-P model of a fluid is particularly suitable for describing the rheology of dilute polymer solutions (Newtonian solvents containing small amounts of dissolved polymer) as a result of its ability to capture nonlinear effects arising from the finite extensibility of the polymer chains. In extensional flows, these polymer solutions exhibit dramatically different behavior from the corresponding Newtonian solvents alone, notably through the creation of persistent filaments when stretched. By using the technique of capillary thinning to study the dynamics of the thinning process of these filaments, the transient extensional rheology of the fluid can be characterized. We show that under conditions of uniaxial elongational flow, a composite analytic solution can be developed to predict the time evolution of the radius of the filament. Furthermore we derive an analytic expression for the finite time to breakup of the fluid filaments. This breakup time agrees very well with results obtained from full numerical simulations, and both numerics and theory predict an increase in the time to breakup as the finite extensibility parameter b, related to the molecular weight of the polymer, is increased. As b -> infinity, the results converge to an asymptotic result for the breakup time which shows that the breakup time grows as t(break) similar to In(M-w), where M-w is the molecular weight of the dilute polymer solution. (C) 2015 Elsevier B.V. All rights reserved.