The tailor-folding method is proposed to make an all-composite sandwich panel with a carbon fiber reinforced polymer (CFRP) hexagon honeycomb core. Using this method, a CFRP honeycomb core with continuous fibers is fabricated automatically from a continuous plain woven prepreg to reinforce the constraints between adjacent cell walls. The analytical expressions were derived for predicting the stiffness and strength of the CFRP hexagon honeycomb sandwich panels for out-of-plane compression and shear loadings. The corresponding failure mechanism maps were also constructed for estimating the dominant failure mode of the honeycomb core, including elastic buckling and fracture under out-of-plane compressive and shear loadings. Selected geometries of the sandwich panels were tested to illustrate these failure modes, with a reasonable agreement between the analytical predictions and experiments. It was observed that the CFRP hexagon honeycomb exhibit good specific energy absorption ability. The tailor-folding method has the potential to provide new opportunities for lightweight multifunctional honeycombs.