The same happens if a nonartifact creature you control with no +1/+1 counters on it has damage marked on it equal to its toughness minus 1. For example, if a 5/5 creature was dealt 4 damage earlier in the turn and its controller resolves The Flesh Is Weak, that creature will die before the +1/+1 counter is placed on it.

If The Flesh is Weak enters the battlefield while you control a nonartifact creature with 1 toughness and no +1/+1 counters on it, the last ability will cause its toughness to become 0 and it will die before the first ability can put a counter on it.