The inclusive Higgs boson production cross-section is measured in the di-photon and the $ZZ^* \rightarrow 4 \ell $ decay channels using 31.4 and 29.0 fb$^{-1}$ of pp collision data respectively, collected with the ATLAS detector at a centre-of-mass energy of $\sqrt{s}=13.6$ $\text {TeV}$. To reduce the model dependence, the measurement in each channel is restricted to a particle-level phase space that closely matches the channel’s detector-level kinematic selection, and it is corrected for detector effects. These measured fiducial cross-sections are $\sigma _{\textrm{fid},\gamma \gamma } = $ $76^{+14}_{-13}$ fb, and $\sigma _{\textrm{fid},4 \ell } =$ $2.80\, \pm \, 0.74$ fb, in agreement with the corresponding Standard Model predictions of $67.6 \pm 3.7 $ fb and $3.67 \pm 0.19 $ fb. Assuming Standard Model acceptances and branching fractions for the two channels, the fiducial measurements are extrapolated to the full phase space yielding total cross-sections of $\sigma (pp \rightarrow H) = 67^{+12}_{-11}$ pb and $46 \pm 12$ pb at 13.6 $\text {TeV}$ from the di-photon and $ZZ^* \rightarrow 4 \ell $ measurements respectively. The two measurements are combined into a total cross-section measurement of $\sigma (pp \rightarrow H)= 58.2 \pm 8.7$ pb, to be compared with the Standard Model prediction of $\sigma (pp \rightarrow H)_\textrm{SM} = 59.9 \pm 2.6 $ pb.