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A graph $\Gamma$ is called $G$-symmetric if it admits $G$ as a group ofautomorphisms acting transitively on the set of ordered pairs of adjacentvertices. We give a classification of $G$-symmetric graphs $\Gamma$ with$V(\Gamma)$ admitting a nontrivial $G$-invariant partition $\mathcal{B}$ suchthat there is exactly one edge of $\Gamma$ between any two distinct blocks of$\mathcal{B}$. This is achieved by giving a classification of $(G,2)$-point-transitive and $G$-block-transitive designs $\mathcal{D}$ togetherwith $G$-orbits $\Omega$ on the flag set of $\mathcal{D}$ such that $G_{\sigma,L}$ is transitive on $L \setminus \{\sigma\}$ and $L \cap N = \{\sigma\}$ fordistinct $(\sigma, L), (\sigma, N) \in \Omega$, where $G_{\sigma, L}$ is thesetwise stabilizer of $L$ in the stabilizer $G_{\sigma}$ of $\sigma$ in $G$.Along the way we determine all imprimitive blocks of $G_{\sigma}$ on $V\setminus \{\sigma\}$ for every $2$-transitive group $G$ on a set $V$, where$\sigma \in V$.

Vertex-imprimitive symmetric graphs with exactly one edge between any two distinct blocks