We report the discovery with the RXTE/PCA of twin kHz QPOs in Cir X-1. Eleven cases of simultaneous double QPOs occurred, with significances of up to 6.3 and 5.5 sigma and centroid frequencies ranging between approximately 56-225 and 230-500 Hz for the two QPO peaks, respectively, i.e., for the most part at frequencies well below those of other sources. The QPO properties clearly indicate that these double peaks are the kHz QPOs known from low magnetic field neutron stars, and not black-hole high-frequency QPOs, confirming that Cir X-1 is a neutron star. The kHz QPO peak separation varies over a wide range, ~175-340 Hz, and increases with QPO frequency. This is contrary to what is seen in other sources but agrees with predictions of the relativistic precession model and Alfven wave models; beat-frequency models require modification to accommodate this. In other observations single kHz QPOs can be seen down to frequencies as low as ~12 Hz, as well as a strong low-frequency (LF) QPO between 1 and 30 Hz. The relations between the frequencies of the kHz QPOs and the LF QPO are in good agreement with those found previously in Z sources, confirming that Cir X-1 may be a peculiar Z source. We suggest that the low frequencies of the kHz QPOs in Cir X-1 and to a lesser extent in (other) Z sources might be due to a relatively stronger radial inflow to the neutron star than in other kHz QPO sources.