A regular-graph design is a block design for which a pair $\{a,b\}$ of distinct points occurs in $λ+1$ or $λ$ blocks depending on whether $\{a,b\}$ is or is not an edge of a given $δ$-regular graph. Our paper describes a specific construction for regular-graph designs with $λ= 1$ and block size $δ+ 1$. We show that for $δ\in \{2,3\}$, certain necessary conditions for the existence of such a design with $n$ points are sufficient, with two exceptions in each case and two possible exceptions when $δ= 3$. We also construct designs of orders 105 and 117 for connected 4-regular graphs.