We present the results of the first complete unbiased survey of the Galactic plane for 6035-MHz excited-state hydroxyl (ex-OH) masers undertaken as part of the methanol multibeam (MMB) survey. These observations cover the Galactic longitude ranges 186deg<l<60deg including the Galactic Centre. We report the detection of 127 ex-OH masers within the survey region, 47 being new sources. The positions of new detections were determined from interferometric observations with the Australia Telescope Compact Array. We discuss the association of 6035-MHz masers in our survey with the 6668-MHz masers from the MMB Survey, finding 37 likely CH_3_OH-ex-OH maser pairs with physical separations of <=0.03pc and 55 pairings separated by <=0.1pc. Using these we calculate for the first time an ex-OH maser lifetime of between 3.3x10^3^ and 8.3x10^3^yr. We also discuss the variability of the 6035-MHz masers and detection rates of counterpart 6030-MHz ex-OH masers (28 per cent of our sample having detection at both frequencies).