An urn contains black and red balls. Let Z
n
be the proportion of black balls at time n and 0≤L<U≤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 that R
n
=B
n
. Then, under mild conditions, it is shown that Z
n
→a.s.
Z for some random variable Z, and D
n
≔√n(Z
n
-Z)→𝒩(0,σ2) conditionally almost surely (a.s.), where σ2 is a certain random variance. Almost sure conditional convergence means that ℙ(D
n
∈⋅|𝒢
n
)→w 𝒩(0,σ2) a.s., where ℙ(D
n
∈⋅|𝒢
n
) is a regular version of the conditional distribution of D
n
given the past 𝒢
n
. Thus, in particular, one obtains D
n
→𝒩(0,σ2) stably. It is also shown that L<Z<U a.s. and Z has nonatomic distribution.