  Consider the blow-up Y of a weighted projective plane at a point in the open
orbit over a field of characteristic 0. We assume that there exists a curve C
on Y such that C^2<0 and C.E=1, where E is the exceptional curve. In this paper
we give a (very simple) necessary and sufficient condition for finite
generation of the Cox ring of Y (Theorem~1.2). It is an affirmative answer to a
conjecture due to He and Kurano-Nishida.
