A related question on StackOverflow is this one. Also the answer by Haile to this one is there:
Since the universe of strings over any finite alphabet is countable,
every language can be mapped to a subset of the natural numbers. So
you just have to take a Recursively enumerable language wich is not
decidable and map it into a subset of $\{1\}^*$.
For example, in the classic version of the halting problem we
enumerate every turing machine into a binary string; you can now sort
all the turing machines and define a map $f : TM \rightarrow N$ from Turing
machines to integers where $f(TM) = n$ if TM is the nth Turing Machine
in the ordered list of all TM.
Now, the halting problem for Turing Machines coded as unary numbers is
r.e. but not decidable.
Another related question in this one.