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The exact two-spinon part of the dynamic spin structure factor Sxx(Q,ω) for the one-dimensional s = 1/2, XXZ model at T = 0 in the antiferromagnetically ordered phase is calculated using recent advances in the algebraic analysis based on (infinite-dimensional) quantum group symmetries of this model and the related vertex models. The two-spinon excitations form a two-parameter continuum consisting of two partly overlapping sheets in (Q,ω) space. The spectral threshold has a smooth maximum at the Brillouin zone boundary (Q = π/2) and a smooth minimum with a gap at the zone center (Q = 0). The two-spinon density of states has square-root divergences at the lower and upper continuum boundaries. For the two-spinon transition rates, the two regimes 0 < ~Q < Qκ (near the zone center) and Qκ < Q < ~π/2 (near the zone boundary) must be distinguished, where Qκ → 0 in the Heisenberg limit and Qκ → π/2 in the Ising limit. In the regime Qκ < Q < ~π/2, the two-spinon transition rates relevant for Sxx(Q,ω) are finite at the lower boundary of each sheet, decrease monotonically with increasing ω, and approach zero linearly at the upper boundary. The resulting two-spinon part of Sxx(Q,ω) is then square-root divergent at the spectral threshold and vanishes in a square-root cusp at the upper boundary. In the regime 0 < Qκ < ~π/2, in contrast, the two-spinon transition rates have a smooth maximum inside the continuum and vanish linearly at either boundary. In the associated two-spinon line shapes of Sxx(Q,ω), the linear cusps at the continuum boundaries are replaced by square-root cusps. Existing perturbation studies have been unable to capture the physics of the regime Qκ < Q < ~π/2. However, their line-shape predictions for the regime 0 < ~Q < Qκ are in good agreement with the exact results if the anisotropy is very strong. For weak anisotropies, the exact line shapes are more asymmetric.