The results of a search for neutral MSSM Higgs boson production in $pp$ collisions at the LHC at a center-of-mass energy of 7~TeV are presented, based on a data sample corresponding to an integrated luminosity of 36 pb$^{-1}$. We search for decays of the Higgs bosons to tau lepton pairs, using the final states with one identified $e$ or $\mu$ from leptonic tau decay plus a reconstructed hadronic decay of a tau lepton, and also the final state with two leptonic tau decays, and then fully reconstruct the original tau lepton pair kinematics using a novel likelihood algorithm. No excess is observed in the fully reconstructed tau pair invariant mass spectrum. The resulting upper bounds on the Higgs production cross section times branching ratio to tau pairs as a function of the Higgs boson mass yield stringent new bounds in the MSSM parameter space $m_A$ versus $\tan\beta$, extending as low as $\tan\beta = 23$ for $m_A=130$ GeV/$c^2$.

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