The design of a cylindrical hollow-cored Bragg fibre for HE11-like mode propagation is presented. The bulk of the propagating power can be contained in the hollow core, leading to material and leakage loss values of interest in the longer wavelength regions at and beyond 10.6 µm. For example, for a core radius of 1 mm, four cladding layers with alternate indices 1.5 and 2.38, the loss would be approximately 4.10−2dB/m at a wavelength of 10.6 µm. The loss can, in general, be further reduced either by increasing the core radius, the index difference between cladding layers, or the number of these layers.