Hollow-core fibers have demonstrated record performance in applications such as high-power pulse delivery, quantum computing, and sensing. However, their routine use is yet to become reality. A major obstacle is the ability to maintain the polarization state of light over a broad range of wavelengths, while also ensuring single-mode guidance and attenuation that is low enough for practical applications that require only a few meters of fiber length (<1 dB/m). Here we simulated, fabricated and characterized a single-mode birefringent anti-resonant hollow-core fiber. The birefringence was achieved by introducing capillary tubes of different thicknesses, thereby creating reduced symmetry in the structure. The measured group birefringence is in good agreement with the calculated group birefringence from simulations across the fiber guidance band within the telecommunications C-band. At 1550 nm, we measured a group birefringence of 4.4×10-5, which corresponds to a phase birefringence of 2.5×10-5. The measured loss of the fiber was 0.46 dB/m at 1550 nm. The measured polarization extinction ratio of the fiber at 1550 nm was 23.1 dB (25.7 dB) along the x-(y-) polarization axis, relating to an h-parameter of 9.8×10-4 (5.3×10-4).