Is the counting version of 1-in-3 Sat #P-complete?
Yes, the counting version of 1-in-3 Sat is $\#P$-complete. This is stated in "Complexity of Generalized Satisfiability Counting Problems" (Example 3.1), the reference pointed out by Emil in the comment.

Note:
This leads us to the curious question: is the counting version of every NP-complete problem $\#P$-complete? The answers to this question (When does &quot;X is NP-complete&quot; imply &quot;#X is #P-complete&quot;?) suggest so!