Hi,
Rather long after your question, but it can be done directly in the same way Matus did it, or you can simply use the following:

Matus assumed weights Wi which sum to 1. Suppose you have weights Ui, and write V1 = sum of the Ui, and V2 = sum of the Ui^2, consistent with the Wikipedia entry for weighted sample variance.
Then we can put Wi = Ui/V1. 

Now, look at the factor 1 / (1 - sum(Wi^2)), replace the Wi with Ui/V1, multiply top and bottom lines by V1^2 and - voila! - you get V1^2 / { V1^2 - V2 }  .

However, like Matus, I'm wondering when you would ever use such a "weighted sample variance" - see my question as a response to the original post.

I suspect there is much confusion over the different reasons for weighting.

Kathy