We report T-c and Co-59 nuclear quadrupole resonance measurements on the cobalt oxide superconductor NaxCoO2 center dot 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 rate of -0.49 +/- 0.09 K/GPa. At low pressures P <= 0.49 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 and 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 T similar to 0.12T(c), which provides compelling evidence for the presence of line nodes in the superconducting gap function.
