  An urn contains black and red balls. Let $Z_n$ be the proportion of black
balls at time $n$ and $0\leq L<U\leq 1$ random barriers. At each time $n$, a
ball $b_n$ is drawn. If $b_n$ is black and $Z_{n-1}<U$, then $b_n$ is replaced
together with a random number $B_n$ of black balls. If $b_n$ is red and
$Z_{n-1}>L$, then $b_n$ is replaced together with a random number $R_n$ of red
balls. Otherwise, no additional balls are added, and $b_n$ alone is replaced.
In this paper, we assume $R_n=B_n$. Then, under mild conditions, it is shown
that $Z_n\overset{a.s.}\longrightarrow Z$ for some random variable $Z$, and
\begin{gather*}
D_n:=\sqrt{n}\,(Z_n-Z)\longrightarrow\mathcal{N}(0,\sigma^2)\quad\text{conditionally
a.s.} \end{gather*} where $\sigma^2$ is a certain random variance. Almost sure
conditional convergence means that \begin{gather*}
P\bigl(D_n\in\cdot\mid\mathcal{G}_n\bigr)\overset{weakly}\longrightarrow\mathcal{N}(0,\,\sigma^2)\quad\text{a.s.}
\end{gather*} where $P\bigl(D_n\in\cdot\mid\mathcal{G}_n\bigr)$ is a regular
version of the conditional distribution of $D_n$ given the past
$\mathcal{G}_n$. Thus, in particular, one obtains
$D_n\longrightarrow\mathcal{N}(0,\sigma^2)$ stably. It is also shown that
$L<Z<U$ a.s. and $Z$ has non-atomic distribution.
