  For a nilpotent Lie algebra $L$ of dimension $n$ and dim$(L^2)=m$, we find
the upper bound dim$(M(L))\leq {1/2}(n+m-2)(n-m-1)+1$, where $M(L)$ denotes the
Schur multiplier of $L$. In case $m=1$ the equality holds if and only if
$L\cong H(1)\oplus A$, where $A$ is an abelian Lie algebra of dimension $n-3$
and H(1) is the Heisenberg algebra of dimension 3.
