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In contrast to the classical super-Brownian motion (SBM), the SBM (Xϱt) t ≥ 0 in a super-Brownian medium ϱ (constructed in [DF96a]) is known to be persistent in all three dimensions of its non-trivial existence: The full intensity is carried also by all longtime limit points ([DF96a, DF96b, EF96]). Uniqueness of the accumulation point, however, has been shown so far only in dimensions d=1 and d=3 ([DF96a, DF96b]). Here we fill this gap and show that convergence also holds in the critical dimension d=2. We identify the limit as a random multiple of Lebesgue measure. Our main tools are a self-similarity of Xϱ in d=2 and the fact that the medium has "gaps" in the space-time picture. The self-similarity implies that persistent convergence of Xϱt as t → ∞ is equivalent to the absolute continuity of Xϱt at a fixed time t > 0. Absolute continuity however will be obtained via the fact that in absence of the catalytic medium, Xϱ is smoothed according to the heat flow.