We consider a birth–death process {X(t),t≥0} on the positive integers for which the origin is an absorbing state with birth coefficients λ
n
,n≥0, and death coefficients μ
n
,n≥0. If we define A=∑
n=1
∞ 1/λ
n
π
n
and S=∑
n=1
∞ (1/λ
n
π
n
)∑
i=n+1
∞ π
i
, where {π
n
,n≥1} are the potential coefficients, it is a well-known fact (see van Doorn (1991)) that if A=∞ and S<∞, then λ
C
>0 and there is precisely one quasistationary distribution, namely, {a
j
(λ
C
)}, where λ
C
is the decay parameter of {X(t),t≥0} in C={1,2,...} and a
j
(x)≡μ1
-1π
j
xQ
j
(x), j=1,2,.... In this paper we prove that there is a unique quasistationary distribution that attracts all initial distributions supported in C, if and only if the birth–death process {X(t),t≥0} satisfies bothA=∞ and S<∞. That is, for any probability measure M={m
i
, i=1,2,...}, we have lim
t→∞ℙ
M
(X(t)=j∣ T>t)= a
j
(λ
C
), j=1,2,..., where T=inf{t≥0 : X(t)=0} is the extinction time of {X(t),t≥0} if and only if the birth–death process {X(t),t≥0} satisfies both A=∞ and S<∞.