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A new experiment to determine the thermonuclear cross section of the $^{12}\mathrm{C}$($\ensuremath{\alpha},\ensuremath{\gamma}$)$^{16}\mathrm{O}$ reaction has been performed in regular kinematics using an intense \ensuremath{\alpha}-particle beam of up to 340 \ensuremath{\mu}A from the Stuttgart DYNAMITRON. For the first time, a 4\ensuremath{\pi}-germanium-detector setup has been used to measure the angular distribution of the \ensuremath{\gamma} rays at all angles simultaneously. It consisted of an array of nine EUROGAM high-purity Ge detectors in close geometry, actively shielded individually with bismuth germanate crystals. The $^{12}\mathrm{C}$ targets were isotopically enriched by magnetic separation during implantation. The depth profiles of the implanted carbon in the $^{12}\mathrm{C}$ targets were determined by Rutherford backscattering for purposes of cross-section normalization and absolute determination of the $E1$ and $E2\phantom{\rule{0.3em}{0ex}}S$ factors. Angular distributions of the \ensuremath{\gamma} decay to the $^{16}\mathrm{O}$ ground state were measured in the energy range ${E}_{c.m.}=1.30--2.78$ MeV and in the angular range (lab.) 30\ifmmode^\circ\else\textdegree\fi{}--130\ifmmode^\circ\else\textdegree\fi{}. From these distributions, astrophysical $E1$ and $E2\phantom{\rule{0.3em}{0ex}}S$-factor functions vs energy were calculated, both of which are indispensable to the modeling of this reaction and the extrapolation toward lower energies. The separation of the $E1$ and $E2$ capture channels was done both by taking the phase value ${\ensuremath{\phi}}_{12}$ as a free parameter and by fixing it using the results of elastic \ensuremath{\alpha}-particle scattering on $^{12}\mathrm{C}$ in the same energy range.