Let R be a commutative Noetherian ring, I an ideal of R and T be a non-zero I-cofinite R-module with dim(T) ${leq}$ 1. In this paper, for any finitely generated R-module N with support in V(I), we show that the R-modules $Ext^i_R$(T,N) are finitely generated for all integers $i{geq}0$. This immediately implies that if I has dimension one (i.e., dim R/I = 1), then $Ext^i_R$($H^j_I$(M), N) is finitely generated for all integers $i$, $j{geq}0$, and all finitely generated R-modules M and N, with Supp(N) ${subseteq}$ V(I).