The moment the ant takes even one step, the far end will no longer be move away from the ant at 1 km/s, so it will be able to move fast enough to catch up and c

The moment the ant takes even one step, the far end will no longer be move away from the ant at 1 km/s, so it will be able to move fast enough to catch up and cross the rope. After one step, the ant will be somewhere in the middle, and both ends will be moving away from it. (Because when the rope stretches, every point on the rope moves away from every other point on the rope.) If the far end is moving one direction, and the near end is moving the other direction, and the two ends are moving away from each other at 1 km/s, then neither of them can be moving away from the ant at 1 km/s. The ant will then be able to catch up with either end.