We present the results of our $^{13}$C NMR study of the quasi-two-dimensional organic conductor $theta$-(BEDT-TTF)$_2$I$_{3}$ under pressure, which is suggested to be a zero-gap conductor by transport measurements. We found that NMR spin shift is proportional to $T$ and that spin-lattice relaxation rate follows the power law $T^{alpha}$ ($alpha =3 sim 4$), where $T$ is the temperature. This behavior is consistent with the cone-like band dispersion and provides microscopic evidence for the realization of the zero-gap state in the present material under pressure.
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