We report T_c and ^{59}Co nuclear quadrupole resonance (NQR) measurements on the cobalt oxide superconductor Na_{x}CoO_{2}cdot 1.3H_{2}O (T_c=4.8 K) under hydrostatic pressure (P) up to 2.36 GPa. T_c decreases with increasing pressure at an average r
ate of -0.49pm0.09 K/GPa. At low pressures Pleq0.49 GPa, the decrease of T_c is accompanied by a weakening of the spin correlations at a finite wave vector and a reduction of the density of states (DOS) at the Fermi level. At high pressures above 1.93 GPa, however, the decrease of T_c is mainly due to a reduction of the DOS. These results indicate that the electronic/magnetic state of Co is primarily responsible for the superconductivity. The spin-lattice relaxation rate 1/T_1 at P=0.49 GPa shows a T^3 variation below T_c down to Tsim 0.12T_c, which provides compelling evidence for the presence of line nodes in the superconducting gap function.
