We report an accurate measurement of the phase noise of a thermally limited electronic oscillator at 300 K. By thermally limited we mean that the white signal-to-noise ratio of the oscillator is at or near the level generated by the thermal noise of the 50 ohm source resistor. The measurement is devoid of the anti-correlation effect that originates from the common mode power splitter in a cross-spectrum technique. The anti-correlation effect is mitigated by cooling the power splitter to a liquid helium temperature (4 K). The measurements in this paper are the first proof of theoretical claims that additive thermal noise from the splitter can be reduced significantly with cryogenic cooling and this can eliminate any anti-correlated noise introduced by use of the two-channel cross-spectrum technique. We also confirm measurements of partial anti-correlation error of (-1.3 +/- 0.6) dB that agree with theory when the splitter is at liquid nitrogen temperature of 77 K.