It is well known that the result of any phase measurement on an optical mode made using linear optics has an introduced uncertainty in addition to the intrinsic quantum phase uncertainty of the state of the mode. The best previously published technique [H. M. Wiseman and R. B. Killip, Phys. Rev. A 57, 2169 (1998)] is an adaptive technique that introduces a phase variance that scales as n¯−1.5 , where ¯n is the mean photon number of the state. This is far above the minimum intrinsic quantum phase variance of the state, which scales as ¯n − 2 . It has been shown that a lower limit to the phase variance that is introduced scales as ln(¯n ) / n¯ 2 . Here we introduce an adaptive technique that attains this theoretical lower limit.