We consider families of analytic area-preserving maps depending on two pa-rameters: the perturbation strength E and the characteristic exponent h of theorigin. For E=0, these maps are integrable with a separatrix to the origin,whereas they asymptote to flows with homoclinic connections as h-&0+. Forfixed E!=0 and small h, we show that these connections break up. The area ofthe lobes of the resultant turnstile is given asymptotically by E exp(-Pi^2/h)Oª(h),where Oª(h) is an even Gevrey-1 function such that Oª(0)!=0 and the radiusof convergence of its Borel transform is 2Pi^2. As E-&0 the function Oª tendsto an entire function Oº. This function Oº agrees with the one provided by theMelnikov theory, which cannot be applied directly, due to the exponentially smallsize of the lobe area with respect to h.These results are supported by detailed numerical computations; we use anexpensive multiple-precision arithmetic and expand the local invariant curves upto very high order.