  We construct modular invariant partition functions for strings propagating on
non-compact manifolds of G_2 holonomy. Our amplitudes involve a pair of N=1
minimal models M_m, M_{m+2} (m=3,4,...) and are identified as describing
strings on manifolds of G_2 holonomy associated with A_{m-2} type singularity.
It turns out that due to theta function identities our amplitudes may be cast
into a form which contain tricritical Ising model for any m. This is in accord
with the results of Shatashvili and Vafa. We also construct a candidate
partition function for string compactified on a non-compact Spin(7) manifold.
