In the limit of dominant nearest-neighbour hopping and infinitesimal secondand third-nearest-neighbour hopping, the Hubbard model is solvable in the sense that it separates into two models which have previously been analysed in the literature. The solutions are spin-charge separated and involve highly correlated motion of the electrons. Although in general the solutions are too complicated to analyse, there are two quite specific states which are amenable to direct analytical attack: the phase boundary between the quantum paramagnet and the Nagaoka ferromagnet, and the 'dimer' phase where all spins pair up into nearest-neighbour singlets. We have been able to evaluate both (n(k)) and (S_-k.S_k) accurately enough to deduce the qualitative behaviour for the two types of state in question. The second of these correlation functions is directly relevant to neutron-scattering studies. Although the solutions are highly correlated and spin-charge separated, the correlations are reminiscent of the non-interacting 'spinfull' free-electron gas solution, the spin response exhibiting a divergence where the noninteracting susceptibility does, for example.