  A weak magnetic order was found in a non-superconducting bilayered-hydrate
Na$_{x}$CoO$_{2}\cdot y$H$_{2}$O sample by a Co Nuclear Quadrupole Resonance
(NQR) measurement. The nuclear spin-lattice relaxation rate divided by
temperature $1/T_1T$ shows a prominent peak at 5.5 K, below which a Co-NQR peak
splits due to an internal field at the Co site. From analyses of the Co NQR
spectrum at 1.5 K, the internal field is evaluated to be $\sim$ 300 Oe and is
in the $ab$-plane. The magnitude of the internal field suggests that the
ordered moment is as small as $\sim 0.015$ $\mu_B$ using the hyperfine coupling
constant reported previously. It is shown that the NQR frequency $\nu_Q$
correlates with magnetic fluctuations from measurements of NQR spectra and
$1/T_1T$ in various samples. The higher-$\nu_Q$ sample has the stronger
magnetic fluctuations. A possible phase diagram in Na$_{x}$CoO$_{2}\cdot
y$H$_{2}$O is depicted using $T_c$ and $\nu_Q$, in which the crystal distortion
along the c-axis of the tilted CoO$_2$ octahedron is considered to be a
physical parameter. Superconductivity with the highest $T_c$ is seemingly
observed in the vicinity of the magnetic phase, suggesting strongly that the
magnetic fluctuations play an important role for the occurrence of the
superconductivity.
