From the interactions of a deuteron beam in a liquid-hydrogen target at energies near the $d+p\ensuremath{\rightarrow}^{3}\mathrm{He}+\ensuremath{\eta}$ threshold, $\ensuremath{\eta}$ events were selected on the basis of the momentum-analyzed $^{3}\mathrm{He}$ with a background of less than 2%. This opens the way to a $^{3}\mathrm{He}$-tagged $\ensuremath{\eta}$ beam at Saturne, with a possible flux of ${10}^{4}$-${10}^{5}$ $\ensuremath{\eta}'\mathrm{s}$ per pulse. The present data allow an evaluation of the cross section down to less than 0.5 MeV above threshold. The deuteron tensor analyzing power ${t}_{20}$ has also been measured: The small negative value found at threshold (-0.15 \ifmmode\pm\else\textpm\fi{} 0.05) seems incompatible with models based on one-pion exchange.