The Fock–Bargmann–Hartogs domain $$D_{n,m}(\mu )$$ ( $$\mu >0$$ ) in $$\mathbb {C}^{n+m}$$ is defined by the inequality $$\Vert w\Vert ^2<e^{-\mu \Vert z\Vert ^2},$$ where $$(z,w)\in \mathbb {C}^n\times \mathbb {C}^m$$ , which is an unbounded non-hyperbolic domain in $$\mathbb {C}^{n+m}$$ . This paper introduces a Kähler metric $$\alpha g(\mu ;\nu )$$ $$(\alpha >0)$$ on $$D_{n,m}(\mu )$$ , where $$g(\mu ;\nu )$$ is the Kähler metric associated with the Kähler potential $$\Phi (z,w):=\mu \nu {\Vert z\Vert }^{2}-\ln (e^{-\mu {\Vert z\Vert }^{2}}-\Vert w\Vert ^2)$$ ( $$\nu >-1$$ ) on $$D_{n,m}(\mu )$$ . The purpose of this paper is twofold. Firstly, we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space of square integrable holomorphic functions on $$(D_{n,m}(\mu ), g(\mu ;\nu ))$$ with the weight $$\exp \{-\alpha \Phi \}$$ for $$\alpha >0$$ . Secondly, using the explicit expression of the Bergman kernel, we obtain the necessary and sufficient condition for the metric $$\alpha g(\mu ;\nu )$$ $$(\alpha >0)$$ on the domain $$D_{n,m}(\mu )$$ to be a balanced metric. So, we obtain the existence of balanced metrics for a class of Fock–Bargmann–Hartogs domains.