Deep spectroscopic observations of z~6.5 galaxies have revealed a marked decline with increasing redshift in the detectability of Lyman-alpha emission. While this may offer valuable insight into the end of the reionisation process, it presents a fundamental challenge to the detailed spectroscopic study of the many hundreds of photometrically-selected distant sources now being found via deep HST imaging, and particularly those bright sources viewed through foreground lensing clusters. In this paper we demonstrate the validity of a new way forward via the convincing detection of an alternative diagnostic line, CIII]1909, seen in spectroscopic exposures of two star forming galaxies at z=6.029 and 7.213. The former detection is based on a 3.5 hour X-shooter spectrum of a bright (J=25.2) gravitationally-lensed galaxy behind the cluster Abell 383. The latter detection is based on a 4.2 hour MOSFIRE spectra of one of the most distant spectroscopically confirmed galaxies, GN-108036, with J=25.2. Both targets were chosen for their continuum brightness and previously-known redshift (based on Lyman-alpha), ensuring that any CIII] emission would be located in a favorable portion of the near-infrared sky spectrum. We compare our CIII] and Lyman-alpha equivalent widths in the context of those found at z~2 from earlier work and discuss the motivation for using lines other than Lyman-alpha to study galaxies in the reionisation era.