The submillimeter-wave rotational spectra of CO+, (CO+)-C-13 and (CO+)-O-18 in the nu = 0 and 1 vibrational states were measured through a hollow cathode dc discharge in a cryogenic cell cooled to liquid nitrogen temperature. In addition, a few transitions Of the main isotopic species have been measured between 1.1 and 1.3 THz. An updated isotopically invariant fit, including Born-Oppenheimer breakdown corrections,. is presented: the derived set of independent molecular parameters, valid for all the isotopologues of the molecule included in the fit, allows to predict the rotational spectrum with calculated 1 sigma uncertainty of 280 kHz at 2 THz.