In 1975, Wenger [4] determined the fine spectra of $Ces{grave{a}}ro$ operator $C_1$ on c, the space of convergent sequences. In [7], the spectrum of the Rhaly operators on $c_0$ and c, under the assumption that ${lim}limits_{n{ ightarrow}{infty}}(n+1)a_n;=;L;{ eq};0$, has been determined. In this paper the author determine the fine spectra of the Rhaly matrix $R_a$ as an operator on the space c, with the same assumption.