= Solution
The strong band is $0\to1$, while the weak band is the first <vibrational overtone> $0\to2$. Let $w=\widetilde\omega_e$ and $q=\widetilde\omega_ex_e$. Subtracting the <Morse oscillator> levels gives
$$
\begin{aligned}
\widetilde\nu_{01}&=w-2q=440.6\ \mathrm{cm^{-1}},\\
\widetilde\nu_{02}&=2w-6q=877.5\ \mathrm{cm^{-1}}.
\end{aligned}
$$
Twice the fundamental is $881.2\ \mathrm{cm^{-1}}$, slightly above the overtone because the higher vibrational gap is smaller. Eliminate $w$:
$$
2q=2(440.6)-877.5=3.7\ \mathrm{cm^{-1}},
$$
so
$$
\boxed{\widetilde\omega_ex_e=1.85\ \mathrm{cm^{-1}},\qquad
\widetilde\omega_e=444.3\ \mathrm{cm^{-1}}.}
$$
The implied dimensionless anharmonicity constant is $x_e\simeq0.004164$. The overtone's weakness follows from its being forbidden in the leading harmonic/linear-dipole model, rather than a lack of population in its lower state.
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