Boron nitride nanotubes (BNT) and boron nitride cones (BN cones) were produced by thermal annealing of a mixed powder of beta-rhombohedral boron and hexagonal boron nitride (h-BN) at 1200 degrees C under lithium vapor. The BNTs were found to grow from the edge of h-BN particles along one of [10 (1) over bar 0] directions of the particles. The BNTs consist of zigzag-type layers whose [10 (1) over bar 0] direction is parallel to the tube axis. The BN cones with an apex angle of about 20 degrees were discovered. It was revealed that the BN cones were not formed by a helically wound conical BN layer but by a pile of monolayer BN cones.