  For the Calabi-Yau threefolds $X$ constructed by C. Schoen as fiber products
of generic rational elliptic surfaces, we show that the action of the
automorphism group of $X$ on the K\"ahler cone of $X$ has a rationally
polyhedral fundamental domain. The second author has conjectured that this
statement will hold in general, the example presented here being the first
non-trivial case in which the statement has been checked. The conjecture was
motivated by the desire to use a construction of E. Looijenga to compactify
certain moduli spaces which arise in the study of conformal field theory and
``mirror symmetry.''
