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We study the convergence of the numerical path integration method which produces approximate probability density functions for the solutions of stochastic differential equations. This method enjoys several advantages compared to path-wise approximation schemes. We prove that the approximate probability densities converge strongly in $L^1$ to the probability density of the stochastic differential equation, uniformly in any finite time interval. The concept of dissipative operators and semigroup techniques are used to prove the convergence result. On the way we obtain a semigroup generation result that in itself is new for elliptic operators with unbounded coefficients.