For a nilpotent Lie algebra $L$ of dimension $n$ and dim$(L^2)=m(m\geq 1)$,
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.