Differential cross sections for the coherent photoproduction of ${\ensuremath{\pi}}^{0}$ mesons from ${\mathrm{He}}^{3}[{\mathrm{He}}^{3}(\ensuremath{\gamma},{\ensuremath{\pi}}^{0}){\mathrm{He}}^{3}]$ were measured for recoil angles of 20\ifmmode^\circ\else\textdegree\fi{}, 30\ifmmode^\circ\else\textdegree\fi{}, 40\ifmmode^\circ\else\textdegree\fi{}, and 50\ifmmode^\circ\else\textdegree\fi{} and were found to be 2-8 \ensuremath{\mu}b/sr, 5-25 \ensuremath{\mu}b/sr, 17-56 \ensuremath{\mu}b/sr, and 17-84 \ensuremath{\mu}b/sr, respectively. A gaseous ${\mathrm{He}}^{3}$ target at 20.4\ifmmode^\circ\else\textdegree\fi{}K was irradiated by 340-MeV bremsstrahlung, and ${\mathrm{He}}^{3}$ recoils were observed with silicon semiconductor detectors. At least one $\ensuremath{\gamma}$ ray from the pion decay was detected with a photon shower counter. A coincidence triggered an oscilloscope on which all pulses were displayed and photographed. The experimental cross sections are compared with those obtained by various impulse-approximation calculations. Although the calculations depend on the manner of specifying the energy and momentum dependence of the photon-nucleon amplitudes in the ${\mathrm{He}}^{3}$ nucleus, a calculation assuming that the three nucleons were initially at rest accounts for most of the measurements.