Near the ends of their lives, low- and intermediate-mass stars go through a phase of evolution known as the asymptotic giant branch(1,2). This luminous red-giant phase is thought to be terminated by a period of intense mass loss in the form of a superwind(3), which leads to the formation of a planetary nebula. Although the effects of mass loss have been studied extensively in many stars, the duration of this phase is not well constrained, because of uncertainties in the distances, masses, ages, and absolute luminosities of the observed stars. On the other hand, the properties of stars in the globular clusters associated with the Magellanic Clouds are not subject to these uncertainties, and so provide an excellent opportunity for studying mass-loss phenomena in a quantitative way. Here we report the discovery of two infrared stars in Magellanic Cloud globular clusters that are undergoing a period of intense mass loss. Those observations, together with those of a previously discovered infrared star, confirm that asymptotic giant branch stars go through a superwind phase, and constrain the duration of this phase to be about 100,000 years.