Let R be a Krull ring with the unique regular maximal ideal M. We show that R has a regular prime element and reg-$dimR=1{Leftrightarrow}R$ is a factorial ring and reg-$dim(R)=1{Rightarrow}M$ is invertible ${Leftrightarrow}R{varsubsetneq}[R:M]{Leftrightarrow}M$ is divisorial ${Leftrightarrow}$ reg-$htM=1{Rightarrow}R$ is a rank one discrete valuation ring. We also show that if M is generated by regular elements, then R is a rank one discrete valuation ring ${Rightarrow}$ R is a factorial ring and reg-dim(R)=1.