As stated by many respondents above, it is perfectly correct to use the double possessive. Consider this slightly altered example:
This is my necklace.
This is a necklace of mine.
Both are correct and, for all intents and purposes, they mean the same thing. But you cannot say "This is a necklace of me." It's ungrammatical.
A good example to illustrate why the double negative is necessary in English:
1. This is a photograph of me.
2. This is a photograph of mine.
3. This is my photograph.
The first example is clear: the person in the photograph is me. There is no indication of ownership of the photograph.
The second example shows clearly the ownership of the photograph (mine), but who's in the photograph is not so obvious. It may well be me in the photograph, but the photograph certainly belongs to me.
The third example almost certainly indicates ownership by me, but may also indicate that the photograph is of me. That has to do with the nature of a photograph (as opposed to a necklace or, say, a house). In practice, a necklace or a house cannot be made of “me”, but a photograph can have me as its image. In other words, no one will assume that “This is my necklace” or “This is my house” means that I am the substance of which the necklace or the house is made. At least not in a literal sense.
This is why machines will not soon replace writers and editors. Language is a beautifully complex enterprise.
But back to the question of correct or not: of course people use "incorrect" constructions every day, but that's common usage not writing. It's like clothes. Jeans are comfortable and friendly for everyday wear, but when you get married you want to dress it up a bit.
Writers can and should bend/break grammatical rules, depending on the situation, but only if they know the rules first.