  The K\"ahler rank of compact complex surfaces was introduced by Harvey and
Lawson in their 1983 paper on K\"ahler manifolds as a measure of
``k\"alerianity''. Here we give a partial classification of compact complex
surfaces of K\"ahler rank 1. These are either elliptic surfaces, or Hopf
surfaces, or they admit a holomorphic foliation of a very special type. As a
consequence we give an affirmative answer to the question raised by Harvey and
Lawson whether the K\"ahler rank is a birational invariant.
