It is known that the problem of determining, given a planar graph G and an integer m, whether there exists a congestion-1 embedding of G into an m×k-grid is NP-complete for a fixed integer k≥3. It is also known that the problem for k=3 is NP-complete even if G is restricted to an acyclic graph. The complexity of the problem for k=2 was left open. In this paper, we show that for k=2, the problem can be solved in polynomial time if G is restricted to a tree, while the problem is NP-complete even if G is restricted to an acyclic graph.