Assembly for manipulating a plurality of straws for packaging animal semen each having a predetermined diameter and a predetermined length, comprising a freezing rack having a rim and a bottom surrounding a central free space, said bottom comprising two toothed walls (11, 25) which extend longitudinally along said central space, each toothed wall (11, 25) having a set of toothing at its apex comprising a succession of teeth and hollows, the sets of toothing of said toothed walls (11, 25) being identical with each hollow delimited by a surface having a curvature according to said predetermined diameter; wherein - in said freezing rack said bottom comprises a bottom wall in the form of a rectangular frame surrounding the central free space with two transverse sections (7, 8) and two longitudinal sections (9, 10), said toothed walls (11, 25) extending transversely to the bottom wall and internally bordering said longitudinal sections (9, 10), said rim extending transversely to the bottom wall and comprising two longitudinal walls (12, 13) externally bordering said longitudinal sections (9, 10) of the bottom wall and two transverse walls (14, 15) externally bordering said transverse sections (7, 8) of the bottom wall, said longitudinal walls (12, 13) each comprising at their apex a gutter portion; - said assembly further comprises a transfer device for straws comprising two lateral skids and a collector for straws that is disposed between the two lateral skids, said freezing rack and said transfer device being configured to have a first extreme position in which said transfer device is disposed with each lateral skid bearing on a said gutter portion while said collector is located over a said transverse section (7, 8) of said bottom wall, to have a second extreme position in which said transfer device is disposed with each lateral skid bearing on a said gutter portion while said collector is located over the other said transverse section (7, 8) of said bottom wall, and such that by sliding the transfer device from the first extreme position to the second extreme position with said skids sliding on said gutter portions, said plurality of straws initially disposed on said rack is collected in said collector of said transfer device. - in said freezing rack said bottom comprises a bottom wall in the form of a rectangular frame surrounding the central free space with two transverse sections (7, 8) and two longitudinal sections (9, 10), said toothed walls (11, 25) extending transversely to the bottom wall and internally bordering said longitudinal sections (9, 10), said rim extending transversely to the bottom wall and comprising two longitudinal walls (12, 13) externally bordering said longitudinal sections (9, 10) of the bottom wall and two transverse walls (14, 15) externally bordering said transverse sections (7, 8) of the bottom wall, said longitudinal walls (12, 13) each comprising at their apex a gutter portion; - said assembly further comprises a transfer device for straws comprising two lateral skids and a collector for straws that is disposed between the two lateral skids, said freezing rack and said transfer device being configured to have a first extreme position in which said transfer device is disposed with each lateral skid bearing on a said gutter portion while said collector is located over a said transverse section (7, 8) of said bottom wall, to have a second extreme position in which said transfer device is disposed with each lateral skid bearing on a said gutter portion while said collector is located over the other said transverse section (7, 8) of said bottom wall, and such that by sliding the transfer device from the first extreme position to the second extreme position with said skids sliding on said gutter portions, said plurality of straws initially disposed on said rack is collected in said collector of said transfer device.