The Truncated Moment Problem seeks conditions on an n-dimensional multisequence of degree $m$, $y \equiv (y_i)_{|i| ≤ m}$, such that there exists a positive Borel measure $\mu$ on $\mathbb{R}^n$ satisfying $y_i = \int \xi d \mu \, (|i| ≤ m)$ (where $x = (x_1, \ldots, x_n)$, $i = (i_1, \ldots, i_n)$). In previous work we associated to $y$ an algebraic variety in $\mathbb{R}^n$ , the core variety $V = V(y)$, and showed that if $V$ is nonempty, then the Riesz functional $L$ corresponding to $y$ is strictly V-positive, i.e., if $p(x) := \Sigma a_i x_i \, (|i| ≤ m)$ is nonnegative on $V$, and $p|_V$ is not identically $0$, then $L(p) := \Sigma a_i y_i > 0$. In current work with G. Blekherman, we prove that if $L$ is strictly $K$-positive for any closed subset $K$ of $\mathbb{R}^n$, then $y$ has a representing measure $\mu$ (as above) whose support is contained in $K$. As a consequence, we prove that $y$ has a representing measure if and only if $V(y)$ is nonempty, in which case $V(y)$ coincides with the union of the supports of all representing measures. As a corollary, we obtain a new proof of the Bayer-Teichmann Theorem on multivariable cubature.