  For a block $B$ of a finite group we prove that $k(B)\le(\det
C-1)/l(B)+l(B)\le\det C$ where $k(B)$ (respectively $l(B)$) is the number of
irreducible ordinary (respectively Brauer) characters of $B$, and $C$ is the
Cartan matrix of $B$. As an application, we show that Brauer's
$k(B)$-Conjecture holds for every block with abelian defect group $D$ and
inertial quotient $T$ provided there exists an element $u\in D$ such that
$C_T(u)$ acts freely on $D/<u>$. This gives a new proof of Brauer's Conjecture
for abelian defect groups of rank at most $2$. We also prove the conjecture in
case $l(B)\le 3$.
