We consider the orthogonal polynomial pn(z) with respect to the planar measure supported on the whole complex plane (Formula presented.) where dA is the Lebesgue measure of the plane, N is a positive constant, {c₁, …, c_ν} are nonzero real numbers greater than −1 and (Formula presented.) are distinct points inside the unit disk. In the scaling limit when n/N = 1 and n → ∞ we obtain the strong asymptotics of the polynomial p_n(z). We show that the support of the roots converges to what we call the “multiple Szegő curve,” a certain connected curve having ν + 1 components in its complement. We apply the nonlinear steepest descent method [9,10] on the matrix Riemann-Hilbert problem of size (ν + 1) × (ν + 1) posed in [22].