  A search for proton decay into three charged leptons has been performed by
using 0.37$\,$Mton$\cdot$years of data collected in Super-Kamiokande. All
possible combinations of electrons, muons and their anti-particles consistent
with charge conservation were considered as decay modes. No significant excess
of events has been found over the background, and lower limits on the proton
lifetime divided by the branching ratio have been obtained. The limits range
between $9.2\times10^{33}$ to $3.4\times10^{34}$ years at 90$\,$% confidence
level, improving by more than an order of magnitude upon limits from previous
experiments. A first limit has been set for the $p\rightarrow\mu^-e^+e^+$ mode.
