We present {\it ab initio} and self-consistent tight-binding calculations on the band structure of single wall semiconducting carbon nanotubes with high degrees (up to 25 %) of boron substitution. Besides a lowering of the Fermi energy into the valence band, a regular, periodic distribution of the p-dopants leads to the formation of a dispersive ``acceptor''-like band in the band gap of the undoped tube. This comes from the superposition of acceptor levels at the boron atoms with the delocalized carbon $π$-orbitals. Irregular (random) boron-doping leads to a high concentration of hybrids of acceptor and unoccupied carbon states above the Fermi edge.

4 pages, 2 figures