  We study weak amenability of central Beurling algebras $ZL^1(G,\omega)$. The
investigation is a natural extension of the known work on the commutative
Beurling algebra $L^1(G,\omega)$. For [FC]$^-$ groups we establish a necessary
condition and for [FD]$^-$ groups we give sufficient conditions for the weak
amenability of $Z\L1o$. For a compactly generated [FC]$^-$ group with the
polynomial weight $\omega_\alpha(x) = (1 + |x|)^\alpha$, we prove that
$ZL^1(G,\omega_\alpha)$ is weakly amenable if and only if $\alpha < 1/2$.
