We investigate the collapse of an axisymmetric cavity or bubble inside a fluid of small viscosity, like water. Any effects of the gas inside the cavity as well as of the fluid viscosity are neglected. Using a slender-body description, we compute the local scaling exponent alpha=dlnh(0)/dlnt(') of the minimum radius h(0) of the cavity, where t(') is the time from collapse. The exponent alpha very slowly approaches a universal value according to alpha=1/2+1/[4 root(-ln(t(')))]. Thus, as observed in a number of recent experiments, the scaling can easily be interpreted as evidence of a single nontrivial scaling exponent. Our predictions are confirmed by numerical simulations.