An algebraic method for rotational energies at a given vibrational state (AMr(v)) is proposed in this study in order to obtain unknown high-lying rovibrational energies. Applications of this method to the ground electronic state X1Σ+ of CO and the excited state C1Σ+ of 39K7Li molecules show the following: (1) the AMr(v) can give the rational upper limit $$\overline J $$ J ¯ of a rotational quantum number of a diatomic electronic state; (2) the full AMr(v) rovibrational energies {EυJ}υ} of given vibrational states not only reproduce all known experimental data excellently but also predict precisely the values of all high-lying rovibrational energies, which may not be available experimentally.