The new action will stay in the Actions palette even after Photoshop is closed and reopened. However, the action is not saved, so if you select the Replace Actions option from the Actions palette menu, you’ll lose your action. You can also lose it if Photoshop crashes. For this reason, you should save your actions. Click the My Personal Actions set name; then open the Actions palette menu, choose Save Actions, and save the action set to disk. You can’t save individual actions, only action sets—but you can create a set with only one action in it, if you want.
If your action doesn’t appear in the Actions palette, you can load it by loading the set in which the action is saved.