  For a stationary and axisymmetric spacetime, the vacuum Einstein field
equations reduce to a single nonlinear PDE in two dimensions called the Ernst
equation. By solving this equation with a {\it Dirichlet} boundary condition
imposed along the disk, Neugebauer and Meinel in the 1990s famously derived an
explicit expression for the spacetime metric corresponding to the
Bardeen-Wagoner uniformly rotating disk of dust. In this paper, we consider a
similar boundary value problem for a rotating disk in which a {\it Neumann}
boundary condition is imposed along the disk instead of a Dirichlet condition.
Using the integrable structure of the Ernst equation, we are able to reduce the
problem to a Riemann-Hilbert problem on a genus one Riemann surface. By solving
this Riemann-Hilbert problem in terms of theta functions, we obtain an explicit
expression for the Ernst potential. Finally, a Riemann surface degeneration
argument leads to an expression for the associated spacetime metric.
