  Let $R$ be a commutative ring with identity. In this paper, we introduce the
concept of weakly $1$-absorbing prime ideals which is a generalization of
weakly prime ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing
prime if for all nonunit elements $a,b,c \in R$ such that $0\neq abc \in I$,
then either $ab \in I$ or $c \in I$. A number of results concerning weakly
$1$-absorbing prime ideals and examples of weakly $1$-absorbing prime ideals
are given. It is proved that if $I$ is a weakly $1$-absorbing prime ideal of a
ring $R$ and $0 \neq I_1I_2I_3 \subseteq I$ for some ideals $I_1, I_2, I_3$ of
$R$ such that $I$ is free triple-zero with respect to $I_1I_2I_3$, then $
I_1I_2 \subseteq I$ or $I_3\subseteq I$. Among other things, it is shown that
if $I$ is a weakly $1$-absorbing prime ideal of $R$ that is not $1$-absorbing
prime, then $I^3 = 0$. Moreover, weakly $1$-absorbing prime ideals of PID's and
Dedekind domains are characterized. Finally, we investigate commutative rings
with the property that all proper ideals are weakly $1$-absorbing primes.
