The Eb (eleventh fret) on my low E string is dead. It gives off a horrible buzz when I play it, while every other position on the guitar is fine. Is there any way to remedy the situation without having to drastically raise the action?

raise the action slightly?
Gibson SG Standard
Ibanez S2170FB
Peavey JSX
Marshall 1960A
Is your 12th fret popping out of the neck? If so, just bring it to your local guitar shop and get it tapped back in.
Dean MS STD V w/ Dimebucker
Dean ML '79
Jackson DKMGT
Randall RG75
Peavey Bandit 112
Boss NS-2
the 11th fret is lower than the rest. get a fret dress. if you get no buzz on the 12th fret and no buzz on the 10th fret, then its a low fret. if you got fret buzz on the frets around it then you have a high fret.

and its likely only on the high e string because the sides of the frets are most over-looked when crowning/levelling.
LP Addict, it's the low E not the high E. I get no buzz on frets 10 and 12 so I suppose the fret is low. What does a fret dress entail?
SO, if it's only one one fret one string, as the eleventh fret on all the other strings are fine, what exact;y should I do to remedy this?
well, you need a fret dress, simple as that. its pretty pricey to do one at home. best to take it to someone who knows what they are doing. to be safe, check the 12th fret, make sure the fret is pushed into the fret slot all the way.