= Solution
Within the stipulated localized, one-moment-per-atom <spin> model, the number density follows directly from mass density and molar mass:
$$
n=\frac{\rho N_A}{M_{\mathrm{mol}}}
=\frac{8908}{0.058693}N_A
=9.1400\times10^{28}\,\mathrm{m^{-3}}.
$$
The <face-centered cubic lattice> is consistent with four atoms per conventional <unit cell>; no additional factor four is needed once $n$ has been obtained from the density. The <Curie constant> is $C=0.71549\,\mathrm K$. Identifying the model's $\theta=\lambda C$ with the specified <Curie temperature> gives
$$
\boxed{\lambda=\frac{627\,\mathrm K}{0.71549\,\mathrm K}=8.7632\times10^2.}
$$
At very low <temperature>, a fully ordered single domain has $M_s=n\mu_B=8.4764\times10^5\,\mathrm{A\,m^{-1}}$. Its <Weiss molecular field> is
$$
\boxed{B_{\mathrm{eff}}=\lambda\mu_0M_s
=\frac{k_BT_C}{\mu_B}=9.3343\times10^2\,\mathrm T.}
$$
The cancellation of $n$ in the middle expression provides a useful check. The ordinary magnetostatic scale is only $\mu_0M_s\simeq1.065\,\mathrm T$, smaller by the factor $\lambda$. Thus $B_{\mathrm{eff}}$ is not a plausible ordinary self-generated dipolar <magnetic field> of the nickel moments. It represents the <exchange interaction> energy in field units, not an additional Maxwell field of that magnitude. This does not assert that such large externally generated fields are forbidden. Nickel is an itinerant <ferromagnet>; the imposed one-$\mu_B$-per-atom model is an estimate rather than its measured microscopic moment.
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