  The present paper concerns the derivation of phase-integral quantization
conditions for the two-centre Coulomb problem under the assumption that the two
Coulomb centres are fixed. With this restriction we treat the general
two-centre Coulomb problem according to the phase-integral method, in which one
uses an {\it a priori} unspecified {\it base function}. We consider base
functions containing three unspecified parameters $C, \tilde C$ and $\Lambda$.
When the absolute value of the magnetic quantum number $m$ is not too small, it
is most appropriate to choose $\Lambda=|m|\ne 0$. When, on the other hand,
$|m|$ is sufficiently small, it is most appropriate to choose $\Lambda = 0$.
Arbitrary-order phase-integral quantization conditions are obtained for these
choices of $\Lambda$. The parameters $C$ and $\tilde C$ are determined from the
requirement that the results of the first and the third order of the
phase-integral approximation coincide, which makes the first-order
approximation as good as possible.
  In order to make the paper to some extent self-contained, a short review of
the phase-integral method is given in the Appendix.
