  Given a measure $m$ on the real line or a finite interval, the "cubic string"
is the third order ODE $-\phi'''=zm\phi$ where $z$ is a spectral parameter. If
equipped with Dirichlet-like boundary conditions this is a nonselfadjoint
boundary value problem which has recently been shown to have a connection to
the Degasperis-Procesi nonlinear water wave equation. In this paper we study
the spectral and inverse spectral problem for the case of Neumann-like boundary
conditions which appear in a high-frequency limit of the Degasperis--Procesi
equation. We solve the spectral and inverse spectral problem for the case of
$m$ being a finite positive discrete measure. In particular, explicit
determinantal formulas for the measure $m$ are given. These formulas generalize
Stieltjes' formulas used by Krein in his study of the corresponding second
order ODE $-\phi''=zm\phi$.
