In this paper, we extend previous results on weak approachability in generalized quitting games with vector payoffs to the general case of absorbing games. Specifically, we show that a necessary condition and a different sufficient condition for weak approachability of a convex set, established by Flesch, Laraki, and Perchet, remain valid in the general case. To do so, we extend results on Blackwell approachability to a setup in which stage weights depend on past actions as well as the current action of player 1 (the approaching player). Additionally, we prove that the strategy used to approach the convex set can be defined in blocks of fixed length, and so it has bounded memory and can be implemented by a finite automata.