  Neutron spectroscopy is an invaluable tool for many scientific and industrial
applications, including underground Dark Matter searches. Neutron-induced
backgrounds produced by cosmic ray muons and the cavern radioactivity can mimic
the expected Dark Matter signal. However, existing neutron detection methods
have several drawbacks and limitations, thus measurements remain elusive. A
promising new approach to neutron spectroscopy is the use of a nitrogen-filled
spherical proportional counter that exploits the $^{14}$N(n,$\alpha$)$^{11}$B
and $^{14}$N(n, p)$^{14}$C reactions. This is a safe, inexpensive, effective
and reliable technique. In this work, the latest instrumentation developments
are incorporated in a compact detector operated at the University of Birmingham
(UoB) with high gain at gas pressure up to 1.8\,bar. We demonstrate
spectroscopic measurements of thermalised and fast neutrons respectively from
an $^{241}$Am-$^9$Be source and from the MC40 cyclotron facility at UoB.
Additionally, the detector response to neutrons is simulated using a framework
developed at UoB and compared with the experimental results.
