A hybrid composite and steel rifle barrel assembly relating to bolt action rifles or other firearms, including those that fire rimfire or centerfire ammunition. The inner barrel construction consists of a pre-machined and rifled steel barrel that has been machined down from its original geometry to a much lighter and smaller geometry to achieve a significant weight savings. The metal portion of the barrel that is removed, is replaced by a composite tubular structure, that slips over the machined down steel barrel which is then adhesively bonded to the steel barrel. The composite barrel consists of a plurality of concentric composite material layers including carbon fiber-based uni- directional prepreg and a unique metallic weave that is placed within the composite structure to aid in the thermal transfer of heat extending from the chamber to the muzzle end of the rifle barrel. The metallic weave is positioned so that it is the first layer(s) within the composite structure that make direct contact with the steel barrel. This metallic weave is a continuous weave that extends the full length of the barrel and not only provides a highly thermally conductive layer, but also provides significant longitudinal barrel stiffness. The resin matrix system within the composite structure is a standard epoxy resin that preferably does not contain any type of thermally conductive particulate or filler, to achieve an effective thermal transfer layer. The resulting hybrid composite/steel barrel achieves significant weight reduction compared to an all-steel barrel in addition to increased accuracy. In addition to the weight reduction and accuracy benefits associated with this invention, is that the movement of a bullet associated with shooting through a cold barrel (cold bore) versus a hot barrel is reduced substantially.