Results from a series of long-term numerical integrations of orbits centred on that of C/1995 O1 (Hale-Bopp) are presented. Initially, 33 orbits taken from various sources were integrated in realistic models of the Solar system for various time-scales about the present in the range (-3, +2) Myr and analysed to assess the probability of different dynamical outcomes, such as a Sun-grazing state or the number of orbits since the comet was captured from a long-period orbit in the Oort cloud. Further integrations were performed using an ensemble of 26 orbits more closely clustered about the present orbit of the comet, based on a more accurate method of orbit determination. We find that the ensemble half-life for the comet to be captured or ejected is on the order of 0.5 Myr in the backward integrations and 1.2 Myr in the forward integrations, although a few members of our ensemble were captured during strong encounters with Jupiter within 10 revolutions. Comet Hale-Bopp has a probability p~=0.15 of evolving to a Sun-grazing end-state and, though not a `new' comet (in the sense of being recently captured from the Oort cloud), it may be considered dynamically young.