  Self-gravitating scalar fields with nonminimal coupling to gravity and having
a quartic self-interaction are considered in the domain of outer communications
of a static black hole. It is shown that there is no value of the nonminimal
coupling parameter $\zeta$ for which nontrivial static black hole solutions
exist. This result establishes the correctness of Bekenstein ``no-scalar-hair''
conjecture for quartic self-interactions.
