  In this paper we study non-nilpotent non-Lie Leibniz $\mathbb{F}$-algebras
with one-dimensional derived subalgebra, where $\mathbb{F}$ is a field with
$\operatorname{char}(\mathbb{F}) \neq 2$. We prove that such an algebra is
isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie
Leibniz algebra and an abelian algebra. We denote it by $L_n$, where
$n=\dim_{\mathbb{F}} L_n$. This generalizes the result found in [11], which is
only valid when $\mathbb{F}=\mathbb{C}$. Moreover, we find the Lie algebra of
derivations, its Lie group of automorphisms and the Leibniz algebra of
biderivations of $L_n$. Eventually, we solve the coquecigrue problem for $L_n$
by integrating it into a Lie rack.
