  Six-dimensional N=(1,0) Einstein-Maxwell gauged supergravity is known to
admit a (Minkowski)_4\times S^2 vacuum solution with four-dimensional N=1
supersymmetry. The massless sector comprises a supergravity multiplet, an SU(2)
Yang-Mills vector multiplet, and a scalar multiplet. In this paper it is shown
that, remarkably, the six-dimensional theory admits a fully consistent
dimensional reduction on the 2-sphere, implying that all solutions of the
four-dimensional N=1 supergravity can be lifted back to solutions in six
dimensions. This provides a striking realisation of the idea, first proposed by
Pauli, of obtaining a theory that includes Yang-Mills fields by dimensional
reduction on a coset space. We address the cosmological constant problem within
this model, and find that if the Kaluza-Klein mass scale is taken to be 10^{-3}
eV (as has recently been suggested) then four-dimensional gauge-coupling
constants for bulk fields must be of the order of 10^{-31}. We also suggest a
link between a modification of the model with 3-branes, and a five-dimensional
model based on an S^1/Z_2 orbifold.
