A new phosphor alloy Ca_1+xSr_1-xGa₂S₃F₂: Ce³⁺, Eu²⁺ with x = 0.04 exhibits a full spectrum emission in the entire visible wavelength range from 420 to 700 nm when excited by a LED chip with 400 nm emission. By simply depositing varying thicknesses (amount) of this phosphor powder on a 400 nm excitation LED source, white LEDs with correlated color temperatures (CCT) between 1700 K and 20,000 K and (x,y) chromaticity coordinates lying on the Planckian locus of the CIE chromaticity diagram has been demonstrated. To the best of our knowledge, this is the first report on the demonstration of a single full spectrum phosphor based white LEDs exhibiting the entire gamut of white light color temperature with CRI exceeding 90. Specifically CRI greater than 95 is reported for the CCT range of 2200 K – 8500 K.