The paper examines a higher-order ordinary differential equation of the form $\mathcal{P}[u]\,{:=} \sum_{j,k=0}^{m}D^{j}a_{jk}D^{k}u \,{=}\, \lambda{u}, x\in[0,b)$, where $D\,{=}\,i({d}/{dx})$, and where the coefficients $a_{jk}$, $j,k\in[0,m]$, with $a_{mm}\,{=}\,1$, satisfy certain regularity conditions and are chosen so that the matrix $(a_{jk})$ is hermitean. It is also assumed that $m\,{>}\,1$. More precisely, it is proved, using Paley–Wiener methods, that the corresponding spectral measure determines the equation up to conjugation by a function of modulus 1. The paper also discusses under which additional conditions the spectral measure uniquely determines the coefficients $a_{jk}$, $j,k\in[0,m]$, $j+k\neq{2m}$, as well as $b$ and the boundary conditions at 0 and at $b$ (if any).