Let $H$ be a Cartan subgroup of a semisimple algebraic group $G$ over the complex numbers. The wonderful compactification $\bar H$ of $H$ was introduced and studied by De Concini and Procesi. For the Lie algebra $\fh$ of $H$, we define an analogous compactification $\bar \fh$ of $\fh$, to be referred to as the wonderful compactification of $\fh$. The wonderful compactification of $\fh$ is an example of an "additive toric variety". We establish a bijection between the set of irreducible components of the boundary $\bar \fh - \fh$ of $\fh$ and the set of maximal closed root subsystems of the root system for $(G, H)$ of rank $r - 1$, where $r$ is the dimension of $\fh$. An algorithm based on Borel-de Siebenthal theory that constructs all such root subsystems will be given. We prove that each irreducible component of $\bar \fh - \fh$ is isomorphic to the wonderful compactification of a Lie subalgebra of $\fh$ and is of dimension $r - 1$. In particular, the boundary $\bar \fh - \fh$ is equidimensional. We describe a large subset of the regular locus of $\bar \fh$. As a consequence, we prove that $\bar \fh$ is a normal variety.