= Solution
Interpret the quoted band amplitude as an absolute transit-depth increment $\Delta D=0.0003$, or $300$ parts per million. For a thin atmospheric annulus of extra height $\Delta z$, the <atmospheric spectral-feature amplitude> is
$$
\Delta D=\frac{(R_p+\Delta z)^2-R_p^2}{R_s^2}\simeq\frac{2R_p\Delta z}{R_s^2},\qquad \boxed{\Delta z\simeq\frac{\Delta D}{2D_0}R_p=0.015R_p\simeq1040\,\mathrm{km}.}
$$
Let the <water> band span $N_H$ <atmospheric scale heights>, so $\Delta z=N_HH$. For a nearly <isothermal atmosphere> with constant gravity and <mean molecular weight> $\mu$, $H=k_BT/(\mu m_ug)$. The <temperature from a transmission-band amplitude> is therefore
$$
\boxed{T\simeq\frac{\mu m_ug}{k_B}\frac{\Delta D R_s^2}{2R_pN_H},\qquad \frac{T}{T_0}=\frac{0.015R_p}{N_HH_0}\frac{\mu}{\mu_0}\frac{g}{g_0}.}
$$
Here $T_0,H_0,\mu_0,g_0$ refer to the chosen <Jupiter> reference layer. Assuming $g=g_J\simeq25\,\mathrm{m\,s^{-2}}$, an H/He <exoplanet atmosphere> with $\mu\simeq2.3$, and a clear strong band with $N_H\simeq5$, gives $H\simeq210\,\mathrm{km}$ and \b[$T\simeq1.4\times10^3\,\mathrm K$]. Equivalently, a Jovian reference with $T_0\simeq165\,\mathrm K$ has $H_0\simeq24\,\mathrm{km}$, giving $H/H_0\simeq8.7$.
The assumptions are hydrostatic, approximately isothermal limb gas, constant gravity, negligible cloud truncation, H/He composition, and an effective <opacity> contrast yielding about five scale heights. In an ideal clear limb, $N_H\simeq\log(\kappa_{\rm band}/\kappa_{\rm cont})$ follows from the <slant optical depth of an isothermal atmosphere>. The data fix $N_HH$, not $H$ separately: \b[the temperature is conditional on the band <opacity> contrast and composition]. For $N_H=3$–$7$ with the same composition and gravity, the estimate is about $1000$–$2400\,\mathrm K$.
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