Let $Φ^h(x)$ with $x=(t,y)$ denote the near-critical scaling limit of the planar Ising magnetization field. We take the limit of $Φ^h$ as the spatial coordinate $y$ scales to infinity with $t$ fixed and prove that it is a stationary Gaussian process $X(t)$ whose covariance function is the Laplace transform of a mass spectral measure $ρ$ of the relativistic quantum field theory associated to the Euclidean field $Φ^h$. Our analysis of the small distance/time behavior of the covariance functions of $Φ^h$ and $X(t)$ shows that $ρ$ is finite but has infinite first moment.