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The differential polarizability, due to near-infrared light at 1126 nm, of the 27Al+ 1S0 → 3P0 transition is measured to be Δα = 4πε0 × (1.6 ± 0.5) × 10-31 m3, where Δα = αP - αS is the difference between the excited and ground state polarizabilities. This measurement is combined with experimental oscillator strengths to extrapolate the differential static polarizability of the clock transition as Δα(0) = 4πε0×(1.5-0.5)×10-31 m3. The resulting room temperature blackbody shift of Δυ/υ = -8(3)×10-18 is the lowest known shift of all atomic transitions under consideration for optical frequency standards. A method is presented to estimate the differential static polarizability of an optical transition, from a differential light shift measurement.