  Following the $\alpha$ decay of $^{233}$U, $^{229}$Th recoil ions are shown
to be extracted in a significant amount from the MLL buffer-gas stopping cell.
The produced recoil ions and subsequent daughter nuclei are mass purified with
the help of a customized quadrupole mass spectrometer. The combined extraction
and mass-purification efficiency for $^{229}$Th$^{3+}$ is determined via
MCP-based measurements and via the direct detection of the $^{229}$Th $\alpha$
decay. A large value of $(10\pm2)$\% for the combined extraction and
mass-purification efficiency of $^{229}$Th$^{3+}$ is obtained at a mass
resolution of about 1 u/e. In addition to $^{229}$Th, also other
$\alpha$-recoil ions of the $^{233,232}$U decay chains are addressed.
