A headrest for a motor vehicle seat, said headrest comprising: • an armature having first and second parallel metal rods which are connected to each other by a spacer so as to have, in longitudinal view, the general shape of an inverted U, said rods being intended to be mounted at the top of the back of said seat, • a housing made of moulded plastic material for absorbing energy in the event of a passenger's head impact, said housing having first and a second conduits, resulting from moulding, for respective reception of said first and second sliding rods, so as to allow a height adjustment of said housing, each of said conduits being provided with an upper bearing and a lower bearing both exerting contact tightening on each of said rods in a longitudinal direction, said headrest being characterised in that: • said bearings are integrally moulded with said conduits and have, at the periphery thereof, two opposite flexible tabs in said longitudinal direction, • each of said bearings is provided with a spring for returning said corresponding tabs to said rods, said spring having a general U or C shape, each of the branches of which bears on each of said tabs so as to carry out a pinching of said rods between said tabs to define the sliding force, • said first conduit has one or more bottlenecks extending over a minimum cumulative axial distance (14a, 14b) of 50 mm and having a transverse dimension adjusted to that of said first rod, so as to minimise the transverse movement between said rod and said conduit, • said second conduit is dimensioned so as to ensure an overall transverse clearance (23a, 23b) of at least 2 mm between said conduit and said second rod, so as to avoid, following dimensional variations of said housing resulting from temperature variations, an interference between the two. • an armature having first and second parallel metal rods which are connected to each other by a spacer so as to have, in longitudinal view, the general shape of an inverted U, said rods being intended to be mounted at the top of the back of said seat, • a housing made of moulded plastic material for absorbing energy in the event of a passenger's head impact, said housing having first and a second conduits, resulting from moulding, for respective reception of said first and second sliding rods, so as to allow a height adjustment of said housing, each of said conduits being provided with an upper bearing and a lower bearing both exerting contact tightening on each of said rods in a longitudinal direction, • said bearings are integrally moulded with said conduits and have, at the periphery thereof, two opposite flexible tabs in said longitudinal direction, • each of said bearings is provided with a spring for returning said corresponding tabs to said rods, said spring having a general U or C shape, each of the branches of which bears on each of said tabs so as to carry out a pinching of said rods between said tabs to define the sliding force, • said first conduit has one or more bottlenecks extending over a minimum cumulative axial distance (14a, 14b) of 50 mm and having a transverse dimension adjusted to that of said first rod, so as to minimise the transverse movement between said rod and said conduit, • said second conduit is dimensioned so as to ensure an overall transverse clearance (23a, 23b) of at least 2 mm between said conduit and said second rod, so as to avoid, following dimensional variations of said housing resulting from temperature variations, an interference between the two.