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We prove that if the $(k+n-1)$-secant variety of $X$ has (the expected) dimension $(k+n-1)(n+1)-1<r$ and $X$ is not uniruled by lines, then $X$ is not $k$-weakly defective and hence the $k$-secant variety satisfies identifiability, i.e. a general element of it is in the linear span of a unique $S\\subset X$ with $\\sharp (S) =k$. We apply this result to many Segre-Veronese varieties and to the identifiability of Gaussian mixtures $G_{1,d}$. 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