  The second weight of the Generalized Reed-Muller code of order $d$ over the
finite field with $q$ elements is now known for $d <q$ and $d>(n-1)(q-1)$. In
this paper, we determine the second weight for the other values of $d$ which
are not multiple of $q-1$ plus 1. For the special case $d=a(q-1)+1$ we give an
estimate.
