Motivated by the polynuclear growth model, we consider a Brownian bridge b(t)
with b(\pm T)=0 conditioned to stay above the semicircle c_T(t)=\sqrtT^2-t^2.
In the limit of large T, the fluctuation scale of b(t)-c_T(t) is T^{1/3} and
its time-correlation scale is T^{2/3}. We prove that, in the sense of weak
convergence of path measures, the conditioned Brownian bridge, when properly
rescaled, converges to a stationary diffusion process with a drift explicitly
given in terms of Airy functions. The dependence on the reference point t=\tau
T, \tau\in(-1,1), is only through the second derivative of c_T(t) at t=\tau T.
We also prove a corresponding result where instead of the semicircle the
barrier is a parabola of height T^{\gamma}, \gamma>1/2. The fluctuation scale
is then T^{(2-\gamma)/3}. More general conditioning shapes are briefly
discussed.