  Let $X\subset \mathbb{P}^r$ be an integral and non-degenerate variety. Set
$n:= \dim (X)$. 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 $G_{1,d}$. If $X$ is
the Segre embedding of a multiprojective space we prove identifiability for the
$k$-secant variety (assuming that the $(k+n-1)$-secant variety has dimension
$(k+n-1)(n+1)-1<r$, this is a known result in many cases), beating several
bounds on the identifiability of tensors.
