We demonstrate that twisting a fiber a few turns per meter before it is annealed largely eliminates the residual linear birefringence. This dramatically improves the yield of annealed coils used for current sensing and makes it possible to use fibers that previously had large residual linear birefringence. Twisting the fiber is effective because the residual birefringence, associated with core ellipticity, is reduced to near zero by twisting. A theoretical model of the twisted and annealed fiber current sensor is compared to experimental data. We also show good temperature stability for a sensor made with this new technique.