  A group graded $K$-algebra $A=\bigoplus\limits_{g\in G} A_g$ is called
"locally finite" if $\dim_K A_g < \infty$ for every $g\in G$. We characterise
the weighted graphs $(E,w)$ for which the weighted Leavitt path algebra
$L_K(E,w)$ is locally finite with respect to its standard grading. We also
prove that the locally finite weighted Leavitt path algebras are precisely the
Noetherian ones and that $L_K(E,w)$ is locally finite iff $(E,w)$ is finite and
the Gelfand-Kirillov dimension of $L_K(E,w)$ equals $0$ or $1$. Further it is
shown that a locally finite weighted Leavitt path algebra is isomorphic to a
locally finite Leavitt path algebra and therefore is isomorphic to a finite
direct sum of matrix algebras over $K$ and $K[X,X^{-1}]$.
