Two new alpha emitters 109Xe and 105Te were identified through the observation of the 109Xe --> 105Te --> 101Sn alpha-decay chain. The 109Xe nuclei were produced in the fusion-evaporation reaction 54Fe(58Ni,3n)109Xe and studied using the Recoil Mass Spectrometer at the Holifield Radioactive Ion Beam Facility. Two transitions at Ealpha = 4062 +/- 7 keV and Ealpha = 3918 +/- 9 keV were interpreted as the l = 2 and l = 0 transitions from the 7/2+ ground state in 109Xe (T1/2 = 13 +/- 2 ms) to the 5/2+ ground state and a 7/2+ excited state, located at 150 +/- 13 keV in 105Te. The observation of the subsequent decay of 105Te marks the discovery of the lightest known alpha-decaying nucleus. The measured transition energy Ealpha = 4703 +/- 5 keV and half-life T1/2 = 620 +/- 70 ns were used to determine the reduced alpha-decay width delta2. The ratio delta105Te(2)/delta213Po(2) of approximately 3 indicates a superallowed character of the alpha emission from 105Te.