It has been considered that positrons in pure C-60 are distributed in the interstitial sites between the C-60 molecules. However, it is possible to reside a positron inside a C-60 molecule when the interstitial space between the C-60 molecules become filled with alkali atoms. We report here the positron-annihilation characteristics, lifetime and Doppler-broadening for the annihilation radiation, in K6C60. We have observed that both the lifetime and Doppler-broadening spectrum of K6C60 can be separated into two components. In both measurements, the value of one component is close to the value observed for positron annihilation in pure C-60, where a positron annihilates with an electron outside of a C-60 molecule. On the other hand, we have observed for K6C60 the short lifetime (197 ps), and the extremely small Doppler-broadening on the annihilation spectrum corresponding to the mean electron momentum distribution of 0.95 x 10(-3) m(0)c [full width at half maximum (FWHM)]. These components are considered to reflect positron annihilation inside a C-60 molecule. These demonstrations will lead to further applications to spectroscopy inside a C-60 molecule.