  We have searched for lepton-flavor-violating $\tau$ decays with a
pseudoscalar meson ($\eta$, $\eta'$ {and} $\pi^0$) using a data sample of 401
fb$^{-1}$ collected with the Belle detector at the KEKB asymmetric-energy
$e^+e^-$ collider. No evidence for these decays is found and we set the
following upper limits on the branching fractions: ${\cal{B}}(\tau^-\to
e^-\eta) < 9.2\times 10^{-8}$, ${\cal{B}}(\tau^-\to \mu^-\eta) < 6.5\times
10^{-8}$, ${\cal{B}}(\tau^-\to e^-\eta') < 1.6\times 10^{-7}$,
${\cal{B}}(\tau^-\to \mu^-\eta') < 1.3\times 10^{-7}$, ${\cal{B}}(\tau^-\to
e^-\pi^0) < 8.0\times 10^{-8}$ and ${\cal{B}}(\tau^-\to \mu^-\pi^0) < 1.2\times
10^{-7}$ at the 90% confidence level. These results improve our previously
published upper limits by factors from 2.3 to 6.3.
