Union of R.E. and Non R.E. language
The language $L = L_1 \cup L_2$ consists of all pairs $\langle M,w \rangle$ where $M$ is a description of a Turing machine and $w$ is a word. This language is not necessarily regular (it depends on the exact encoding), but it is certainly decidable. All you have to do, given an input, is to decide whether it is of the form $\langle M,w \rangle$.

Note that deciding $L$ bears no relation to the halting problem. My description of $L$ doesn't refer to whether $M$ halts on $w$ or not. Here is a similar example. Let $f$ be an uncomputable integer-valued function. The function $g := 1-f$ is uncomputable as well, yet $f + g$ is computable.