Precision measurements of slow neutron cross sections with atoms have several scientific applications. In particular, the n−$^4$ He s -wave scattering length is important to know both for helping to constrain the nuclear three-body interaction and for the proper interpretation of several ongoing slow neutron experiments searching for other types of neutron-atom interactions. We present new measurements of the ratios of the neutron differential scattering cross sections of the noble gases He, Ar, Kr, and Xe to Ne. All gases used were of natural isotopic abundance. These measurements were performed using a recently developed neutron-scattering apparatus for gas samples located on a pulsed slow neutron beamline which was designed to search for possible exotic neutron-atom interactions and employs both neutron time of flight information and a position-sensitive neutron detector for scattering event reconstruction. We found agreement with the literature values of scattering cross sections inferred from Ar/Ne, Kr/Ne, and Xe/Ne differential cross-section ratios over the $q$ range of 1–7 nm$^{−1}$. However, for the case of He/Ne we find that the cross section inferred differs by 11.3% ( 7.6 σ ) from previously reported values inferred from neutron phase-shift measurements, but is in reasonable agreement with values from other measurements. The very large discrepancy in the He/Ne ratio calls for a new precision measurement of the n−$^4$ He scattering length using neutron interferometry.