Let τΩ denote the lifetime of Brownian motion in an open connected set Ω ⊂ R. We obtain the asymptotic behaviour of the expected lifetime Ex[τΩ] as y → x, where the Brownian motion is conditioned to start at x and to exit Ω\{y} at {y}. 2000 Mathematics Subject Classification: 60J65, 58J35, 35K20.