If the character has become stable because of aid from another character (such as a Heal check or magical healing), then the character no longer loses hit points. He has a 10% chance each hour of becoming conscious and disabled (even though his hit points are still negative).
If the character became stable on his own and hasn't had help, he is still at risk of losing hit points. Each hour, he has a 10% chance of becoming conscious and disabled. Otherwise he loses 1 hit point.