Here's another one. a little strange, since the quadrature rule involve the value of line integrals, but I guess you can write those using a separate quadrature rule...

Numerical integration over a disc. A new Gaussian quadrature formula

Borislav Bojanov and Guergana Petrova

http://www.springerlink.com/content/xlmv5x2k9uxqlumk/

" We construct a quadrature formula for integration on the unit disc which is based on line integrals over distinct chords in the disc and integrates exactly all polynomials in two variables of total degree . "

Cheers.

Here's another one. a little strange, since the quadrature rule involve the value of line integrals, but I guess you can write those using a separate quadrature rule...

Bojanov, Borislav; Petrova, Guergana, Numerical integration over a disc. A new Gaussian quadrature formula, Numer. Math. 80, No. 1, 39–59 (1998). ZBL0911.65015.

We construct a quadrature formula for integration on the unit disc which is based on line integrals over $n$ distinct chords in the disc and integrates exactly all polynomials in two variables of total degree $2n - 1$.

Cheers.