The Cherenkov Telescope Array (CTA) project aims to create the next generation Very High-Energy (VHE) gamma-ray telescope array. It will be devoted to the observation of gamma rays from 20 GeV to above 100 TeV. Because of this wide energy band, three classes of telescopes, associated with different energy ranges and different mirror sizes, are defined. The Small Size Telescopes (SSTs) are associated with the highest energy range. Seventy of these telescopes are foreseen on the Southern site of the CTA. The large number of telescopes constrains their mechanical structure because easy maintenance and reduced cost per telescope are needed. Moreover, of course, the design shall fulfill the required performance and lifetime in the environment conditions of the site. The Observatoire de Paris started design studies in 2011 of the mechanical structure of the GCT (Gamma-ray Cherenkov Telescope), a four-meter prototype telescope for the SSTs of CTA, from optical and preliminary mechanical designs made by the University of Durham. At the end of 2014 these studies finally resulted in a lightweight ( 8 tons) and stiff design. This structure was based on the dual-mirror Schwarzschild-Couder (SC) optical design, which is an interesting and innovative alternative to the one-mirror Davies-Cotton design commonly used in ground-based Cherenkov astronomy. The benefits of such a design are many since it enables a compact structure, lightweight camera and a good angular resolution across the entire field-of-view. The mechanical structure was assembled on the Meudon site of the Observatoire de Paris in spring 2015. The secondary mirror, panels of the primary mirror and the Telescope Control System were successfully implemented afterwards leading now to a fully operational telescope. This paper focuses on the mechanics of the telescope prototype. It describes the mechanical structure and presents its performance identified from computations or direct measurements. Upgrades of the design in the context of the preproduction and the large scale CTA production are also discussed.