I guess I'm getting (too) old but I just can't wrap my head around this.

Imagine a mylar party balloon that's filled with helium and which is attached to a 10 foot string. At the end of the string is a weight which is resting on the floor. At first of course the balloon will float at the end of the string, 10 feet in the air. Gradually however the helium will leak out and the balloon will over a period of days or weeks slowly sink towards the floor. I know that as the helium leaks out the balloon becomes heavier than the volume of air it displaces so it sinks. I'm probably going to feel silly asking this but why doesn't it fall all the way to the floor the moment there's not enough helium to keep it at 10 feet high?