  We prove that there are no Shimura-Teichm\"uller curves generated by genus
five translation surfaces, thereby completing the classification of
Shimura-Teichm\"uller curves in general. This was conjectured by M\"oller in
his original work introducing Shimura-Teichm\"uller curves. Moreover, the
property of being a Shimura-Teichm\"uller curve is equivalent to having
completely degenerate Kontsevich-Zorich spectrum.
  The main new ingredient comes from the work of Hu and the second named
author, which facilitates calculations of higher order terms in the period
matrix with respect to plumbing coordinates. A large computer search is
implemented to exclude the remaining cases, which must be performed in a very
specific way to be computationally feasible.
