  We present the results of a search for neutral Higgs bosons produced in
association with $b$ quarks in $p\bar{p}\to b\bar{b} \phi\to b\bar{b}b\bar{b}$
final states with $91 \pm 7$ pb$^{-1}$ of $p\bar{p}$ collisions at
$\sqrt{s}=1.8$ TeV recorded by the Collider Detector at Fermilab. We find no
evidence of such a signal and the data is interpreted in the context of the
neutral Higgs sector of the Minimal Supersymmetric extension of the Standard
Model. With basic parameter choices for the supersymmetric scale and the stop
quark mixing, we derive 95% C.L. lower mass limits for neutral Higgs bosons for
$\tb$ values in excess of 35.
