  We investigate a new lattice of generalised non-crossing partitions,
constructed using the geometry of the complex reflection group $G(e,e,r)$. For
the particular case $e=2$ (resp. $r=2$), our lattice coincides with the lattice
of simple elements for the type $D_n$ (resp. $I_2(e)$) dual braid monoid. Using
this lattice, we construct a Garside structure for the braid group $B(e,e,r)$.
As a corollary, one may solve the word and conjugacy problems in this group.
