Motivated by applications in the control theory of infinite-dimensional systems, several authors have investigated the boundedness of the Laplace transform $\mathcal{L}: L^p(0, \infty) \rightarrow L^q( \mathbb{C}_+, \mu)$ in terms of properties of the measure $\mu$, with the case $q=p=2$ being the classical Carleson Embedding Theorem. We will cover the case $p >2$, thus completing the picture to all $p,q$ with $1/p + 1/q \le 1$. In case $p >2$, the Laplace-Carleson Embedding Theorem can also be seen as an appropriate replacement of the Hausdorff-Young inequality. This is joint work with Eskil Rydhe (Lund).