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A $L^2$ Riemannian metric $G_P$ is given on the space of piecewise geodesic paths $H_P(M)$ adapted to the partition $P$ of $[0,1]$, whence a finite-dimensional approximation of Wiener measure is developed. It is proved that, as $mesh(P) \\to 0$, the approximate Wiener measure converges in a $L^1$ sense to the measure $e^{-\\frac{2 + \\sqrt{3}}{20\\sqrt{3}} \\int_0^1 Scal(\\sigma(s)) ds} d\\nu(\\sigma)$ on the Wiener space $W(M)$ with Wiener measure $\\nu$. 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