Measurements of the surface tension of supercooled water down to -25 degrees C have been reported recently (Hruby et al. J. Phys. Chem. Lett. 2014, 5, 425-428). These experiments did not show any anomalous temperature dependence of the surface tension of supercooled water reported by some earlier measurements and molecular simulations. In the present work, this finding is confirmed using a Counterpressure capillary rise method (the counterpressure method) as well as through the use of the classical capillary rise method (the height method). In the counterpressure method, the liquid meniscus inside the vertical capillary tube was kept at a fixed position with an in-house developed helium distribution setup. A preset counterpressure was applied to the liquid meniscus when its temperature changed from a reference temperature (30 degrees C) to the temperature of interest. The magnitude of the counterpressure was adjusted such that the meniscus remained at the same height, thus compensating the change of the surface tension. One advantage of the counterpressure method over the height method consists of avoiding the uncertainty due to a possible variation of the capillary diameter along its length. A second advantage is that the equilibration time due to the capillary flow of the highly viscous supercooled water can be shortened. For both the counterpressure method and the height method, the actual results are relative values of surface tension with respect to the surface tension of water at the reference temperature. The combined relative standard uncertainty of the relative surface tensions is less than or equal to 0.18%. The new data between -26 and +30 degrees C lie close to the IAPWS correlation for the surface tension of ordinary water extrapolated below 0.01 degrees C and do not exhibit any anomalous features.