In a recent publication we have calculated elastic charmonium production in ep collisions employing realistic charmonia wave functions and dipole cross sections and have found good agreement with the data in a wide range of s and Q^2. Using the ingredients from those calculations we calculate exclusive electroproduction of charmonia off nuclei. Here new effects become important, (i) color filtering of the c-cbar pair on its trajectory through nuclear matter, (ii) dependence on the finite lifetime of the c-cbar fluctuation (coherence length) and (iii) gluon shadowing in a nucleus compared to the one in a nucleon. Total coherent and incoherent cross sections for C, Cu and Pb as functions of s and Q^2 are presented together with some differential cross sections. The results can be tested with future electron-nucleus colliders or in peripheral collisions of ultrarelativistic heavy ions.

21 pages of Latex including 14 figures; few misprints are fixed