Based on the feedback given, it seems that unlimited fast healing:1 cannot be balanced without a LA. The advantage of being able to heal fully between combats is too great at lower levels. At mid to high levels, where wands of CLW etc. are very common, this feature becomes less important.
I would eyeball unlimited fast healing 1 as about a LA +2, assuming some kind of buy back. Up till 6th level or so the character would be at an advantage- less so until 9th level. Thereafter fast healing 1 wouldn't make much of a difference. Higher fast healing wouldn't be much better unless it started to make a big difference in combat. Fast healing 5 might be a LA +3, perhaps. Dunno.
Capping the fast healing makes it seem more like normal, natural healing. With rest, everyone gets back 1 hp/level/per day. Using Sigurd's idea of feats, consider allowing a feat that gives 1 hp/level/day of additional healing. This is comparable to Improved Toughness, but maybe on the weak side since it doesn't improve your maximum capacity; the natural healing is also "wasted" if you have a cleric who could easily heal you back to maximum at the end of the day. If you could use it like a personal lay on hands effect it could help you between combats.
I kinda like the idea of someone having fewer maximum hit points (and thus less capable of dealing with one tough fight) but being able to withstand more damage over several encounters on the same day. Thus a Con penalty in exchange for fast healing (even if its capped). And really, a monster's fast healing is usually capped by the length of an encounter- it doesn't fight the characters all day- only for a few rounds. Why shouldn't character's fast healing also be capped? (There's an idea- it works for only 1 encounter/day... too bad that there are no hard rules for determining when an encounter has ended.)
How about this: the character takes a Con penalty of -2, and in exchange gets fast healing 1 for level x level rounds per day. Fast healing can be stopped or started as an immediate action. This exchange of Con for fast healing can be made multiple times; -4 to Con for fast healing 2, etc.. The fast healing doesn't function if the character does not have a Con score.
At first level the hit points are a bit of a wash- your max hit points are 1 less, but you can heal 1 hp of damage 1/day. That allows you to stabilize 1/day automatically. But your fort save is 1 less, and so is your Concentration score, the amount of time you can hold your breath, and so on. No big deal.
At higher levels the hit points might be a concern; attacks that would have left you in positive hit points might now knock you unconscious. But your natural healing will eventually make you a lot less dependent on the party cleric. Of course, some characters will receive multiple heal spells in one combat, not to mention the other ailments that clerics can mend. But still.
A 20th level fighter with 18 Con might have 200 hit points. If he takes fast healing 5 his Con will drop to 8, and he'll have only 100 hit points. He'll be able to heal 5 x 20 x 20 = 2000 hit points per day. Which won't help him if a giant cricals and does him 112 hp in one hit!
Anyway, that's the sort of thing I'm considering right now. People have been leaving some great feedback, and I want to thank you all for it.