  The MZV algebra is the graded algebra over ${\bold Q}$ generated by all
multiple zeta values. The stable derivation algebra is a graded Lie algebra
version of the Grothendieck-Teichm\"{u}ller group. We shall show that there is
a canonical surjective $\bold Q $-linear map from the graded dual vector space
of the stable derivation algebra over $\bold Q$ to the new-zeta space, the
quotient space of the sub-vector space of the MZV algebra whose grade is
greater than 2 by the square of the maximal ideal. As a corollary, we get an
upper-bound for the dimension of the graded piece of the MZV algebra at each
weight in terms of the corresponding dimension of the graded piece of the
stable derivation algebra. If some standard conjectures by Y. Ihara and P.
Deligne concerning the structure of the stable derivation algebra hold, this
will become a bound conjectured in Zagier's talk at 1st European Congress of
Mathematics. Via the stable derivation algebra, we can compare the new-zeta
space with the $l$-adic Galois image Lie algebra which is associated with so
the Galois representation on the pro-$l$ fundamental group of $\bold P ^1_
{\bar{\bold Q}}-{0,1,\infty}$ .
