  The prospects for the discovery of neutral Higgs bosons (phi^0 = H^0, h^0,
A^0) produced with bottom quarks via Higgs decays into muon pairs (pp --> b
b-bar phi^0 --> b b-bar mu mu-bar +X) at the CERN LHC are investigated in the
minimal supersymmetric model. The complete physics background from the
production of b b-bar mu mu-bar, b b-bar W^+ W^- (including t t-bar) and jj mu
mu-bar, j = g, u, d, s, c in the Standard Model is calculated with realistic
acceptance cuts. This discovery mode has a simple production mechanism from gg
--> b b-bar phi^0 with its cross section proportional to 1/cos^2(beta) and
could provide an opportunity to measure tan(beta) and the b b-bar phi^0
couplings. In addition, we compare the associated discovery mode above with the
inclusive discovery channel pp --> phi^0 --> mu mu-bar +X. Promising results
are found for the CP-odd pseudoscalar (A^0) and the heavier CP-even scalar
(H^0) Higgs bosons for tan(beta) equivalent to v_2/v_1 >~ 14 and m_A,m_H <~ 325
GeV.
