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We report results from a study of $B^\pm \rightarrow DK^\pm$ decays followedby $D$ decaying to $CP$~eigenstates, where $D$ indicates a $D^0$ or$\bar{D}^{0}$ meson. These decays are sensitive to theCabibbo-Kobayashi-Maskawa unitarity-triangle angle $\phi_{3}$. The results arebased on a combined analysis of the final data set of $772 \times10^6~B\bar{B}$ pairs collected by the Belle experiment and a data set of $198\times 10^6~B\bar{B}$ pairs collected by the Belle~II experiment, both inelectron-positron collisions at the $\Upsilon(4S)$ resonance. We measure the$CP$ asymmetries to be $\mathcal{ A}_{CP +} =~(+12.5 \pm 5.8 \pm 1.4)\% $ and$\mathcal{ A}_{CP -} =~(-16.7 \pm 5.7 \pm 0.6)\%$, and the ratios of branchingfractions to be $\mathcal{ R}_{CP+}=~1.164 \pm 0.081 \pm 0.036 $ and $\mathcal{R}_{CP-} =~1.151 \pm 0.074 \pm 0.019$. The first contribution to theuncertainties is statistical, and the second is systematic. The asymmetries$\mathcal{A}_{CP +}$ and $\mathcal{A}_{CP -}$ have similar magnitudes andopposite signs; their difference corresponds to 3.5~standard deviations. Fromthese values we calculate 68.3\% confidence intervals of($8.5^{\circ}<\phi_{3}<16.5^{\circ}$) or ($84.5^{\circ}<\phi_{3}<95.5^{\circ}$)or ($163.3^{\circ}<\phi_{3}<171.5^{\circ}$) and $0.321<r_{B}<0.465$.

Measurement of branching-fraction ratios and $CP$ asymmetries in $B^{\pm} \to D_{CP\pm}K^{\pm}$ decays at Belle and Belle II