A design of a polarizing all-glass Bragg fiber with a microstructure core has been proposed for the first time. This design provides suppression of high-order modes and of one of the polarization states of the fundamental mode. The polarizing fiber was fabricated by a new, simple method based on a combination of the modified chemical vapor deposition (MCVD) process and the rod-in-tube technique. The mode field area has been found to be about 870 μm² near λ=1064 nm. The polarization extinction ratio better than 13 dB has been observed over a 33% wavelength range (from 1 to 1.4 μm) after propagation in a 1.7 m fiber piece bent to a radius of 70 cm.