Core modes are spheroidal modes of the Earth's free oscillations whose energy is dominantly partitioned into the inner core as shear energy. We analyzed a total of 87 IDA records of the Earth's free oscillations to detect three core modes 2S2, 6S2 and 7S3 at periods of about 1050, 410, and 320 s, respectively. We used a new autoregressive technique of spectral analysis that determines the periods and Q simultaneously. The Q of these core modes are consistently high, about 2400, 1500, and 1900, respectively. These high values directly demonstrate that the inner core is least dissipative within the Earth for long-period shear waves. Our eigenfrequency data are remarkably consistent with velocity model 1066A [Gilbert and Dziewonski, 1975>. Our attenuation data can be fully explained if the shear wave Q of the inner core is 1500 in the upper 200 km and 3800 in the rest of about 1000 km. ¿ American Geophysical Union 1989