  In heavy nuclei the ratio between local densities of neutrons and protons
increases towards the nuclear periphery. The excess of neutrons is known as the
neutron skin (NS) with a subtle difference ($<0.5$~fm) between the r.m.s radii
of the distributions of neutrons and protons. We show that the presence of NS
in $^{208}$Pb leads to extra spectator neutrons in ultracentral
$^{208}$Pb--$^{208}$Pb collisions at the CERN SPS and LHC. The yields of
spectator neutrons and protons were calculated within a new version of
Abrasion-Ablation Monte Carlo for Colliders model (AAMCC-MST) taking into
account NS and pre-equilibrium clustering of spectator matter. While the
average numbers of spectator neutrons and protons in ultracentral collisions
vary insignificantly, the cross sections of emission of certain numbers of
spectator neutrons in events with 0,1,...5 spectator protons are changed by
50--250\%, depending on the thickness of NS. These cross sections are less
sensitive to other parameters in calculations, and their measurements in the
ALICE experiment at the LHC will make it possible to restrict the existing
variety of neutron density parameterizations in $^{208}$Pb.
