Spheriodal modes of elastic-gravitational oscillation that have an energy partition dominated by shear energy in the inner core are called core modes. Recordings from the IDA gravimeter network of an earthquake in the Tonga region on June 22, 1977 have provided the first unambiguous observations of core modes. They conclusively show that, at low frequencies, the inner core has a very high Q (about 3500). They also tightly constrain the ratio of the inner core radius to the mean shear velocity to be 345s with an uncertainty of 0.5%.