  We determine the Waring ranks of all sextic binary forms with complex
coefficients using a Geometric Invariant Theory approach. Using the five basic
invariants for sextic binary forms, our results give a rapid method to
determine the Waring rank of any given sextic binary form. In particular, we
shed new light on a claim by E. B. Elliott at the end of the 19th century
concerning the binary sextics with Waring rank 3. We show that for binary forms
of arbitrary degree the cactus rank, a.k.a. scheme rank, is determined by the
corresponding Waring rank. Finally we determine the border ranks of all binary
sextics.
