We report the detection of 15 X-ray bursts with RXTE and Swift observations of the peculiar X-ray binary Circinus X-1 (Cir X-1) during its 2010 May X-ray re-brightening. These are the first X-ray bursts observed from the source after the initial discovery by Tennant and collaborators, 25 years ago. By studying their spectral evolution, we firmly identify nine of the bursts as type I (thermonuclear) X-ray bursts. We obtain an arcsecond location of the bursts that confirms once and for all the identification of Cir X-1 as a type I X-ray burst source, and therefore as a low magnetic field accreting neutron star. The first five bursts observed by RXTE are weak and show approximately symmetric light curves, without detectable signs of cooling along the burst decay. We discuss their possible nature. Finally, we explore a scenario to explain why Cir X-1 shows thermonuclear bursts now but not in the past, when it was extensively observed and accreting at a similar rate.