  The Fock-Bargmann-Hartogs domain $D_{n,m}(\mu)$ ($\mu>0$) in
$\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-\mu\|z\|^2},$
where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded
non-hyperbolic domain in $\mathbb{C}^{n+m}$. Recently, Tu-Wang obtained the
rigidity result that proper holomorphic self-mappings of $D_{n,m}(\mu)$ are
automorphisms for $m\geq 2$, and found a counter-example to show that the
rigidity result isn't true for $D_{n,1}(\mu)$. In this article, we obtain a
classification of proper holomorphic mappings between $D_{n,1}(\mu)$ and
$D_{N,1}(\mu)$ with $N<2n$.
