No, it can't be dispelled.
The Dweomerkeeper's Supernatural Spell class feature states (More Divinity web enhancement for Complete Divine, page 2):
At 4th level, the dweomerkeeper is so attuned to the fabric of magic that she can manifest spell effects with almost no effort whatsoever. Once per day as a standard action, she can use any one spell with a casting time of up to 1 standard action as a supernatural ability. The spell chosen must be one that is currently available to the dweomerkeeper (that is, one that she has prepared or that she knows and has a spell slot of the appropriate level available to cast), but she can decide at the moment of casting to use this ability. The spell functions as it normally would and is expended normally, but the dweomerkeeper does not require any components, does not provoke attacks of opportunity, and ignores the target’s spell resistance, just as if she were using a supernatural ability instead of a spell. At every even-numbered level after the 4th, the dweomerkeeper gains one additional use of this ability per day.
And the SRD has this to say about Supernatural Abilities:
Supernatural abilities are magical and go away in an antimagic field but are not subject to spell resistance, counterspells, or to being dispelled by dispel magic. [...]
Emphasis mine in both quotes.
The part that can cast some doubt is:
The spell functions as it normally would and is expended normally, but the dweomerkeeper does not require any components, does not provoke attacks of opportunity, and ignores the target’s spell resistance, just as if she were using a supernatural ability instead of a spell.
Which only lists some of the benefits of the spell becoming a supernatural ability, but that doesn't mean it doesn't gain the unmentioned benefits, like not being able to counterspell or dispel it, because as stated before in the class feature the dweomerkeeper is casting the spell as a supernatural ability, thus gaining all of its benefits.