= Solution
In the <Polyakov path integral>, fix worldsheet diffeomorphism and Weyl symmetry to conformal gauge. The Faddeev-Popov determinant supplies the worldsheet ghosts, and anomaly cancellation selects $D=26$. Insert $m$ integrated closed-string tachyon vertex operators $e^{ip_i\cdot X}$ on the sphere. The zero mode of $X$ gives $\delta^{(26)}(\sum_i p_i)$, while Gaussian contractions give the <Koba-Nielsen factor> $\prod_{j<l}|z_j-z_l|^{\alpha'p_j\cdot p_l}$. Dividing by the conformal Killing group $SL(2,\mathbb C)/\mathbb Z_2$ and using the sphere power of the string coupling gives the displayed amplitude with $g_s^{m-2}$.
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