  We classify finitely generated modules over a class of algebras introduced in
the authors' Ph.D thesis, called complete gentle algebras. These rings
generalise the finite-dimensional gentle algebras introduced by Assem and
Skowro\'{n}ski, in such a way so that the ground field is replaced by any
complete local noetherian ring. Our classification is written in terms of
string and band modules. For the proof we apply the main result from the
authors' thesis, which classifies complexes of projective modules with finitely
generated homogeneous components up to homotopy. In doing so we construct the
resolution of a string or band module, generalising some calculations due to
\c{C}anak\c{c}i, Pauksztello and Schroll.
