We describe the design, data analysis, and basic results of the Giant Metrewave Radio Telescope Cold-Hi AT z approximate to 1 (GMRT-CATz1) survey, a 510 hr upgraded GMRT Hi 21 cm emission survey of galaxies at z = 0.74-1.45 in the DEEP2 survey fields. The GMRT-CAT z1 survey is aimed at characterizing Hi in galaxies during and just after the epoch of peak star formation activity in the universe, a key epoch in galaxy evolution. We obtained high-quality Hi 21 cm spectra for 11,419 blue star-forming galaxies at z = 0.74-1.45, in seven pointings on the DEEP2 subfields. We detect the stacked Hi 21 cm emission signal of the 11,419 star-forming galaxies, which have an average stellar mass of M (*) approximate to 10(10) M (circle dot), at 7.1 sigma statistical significance, obtaining an average Hi mass of M (H i ) = (13.7 +/- 1.9) x 10(9) M (circle dot). This is significantly higher than the average Hi mass of M (H i ) = (3.96 +/- 0.17) x 10(9) M (circle dot) in star-forming galaxies at z approximate to 0 with an identical stellar-mass distribution. We stack the rest-frame 1.4 GHz continuum emission of our 11,419 galaxies to infer an average star formation rate (SFR) of 8.07 +/- 0.82 M (circle dot) yr(-1). Combining our average Hi mass and average SFR estimates yields an Hi depletion timescale of 1.70 +/- 0.29 Gyr, for star-forming galaxies at z approximate to 1, approximate to 3 times lower than that of local galaxies. We thus find that, although main-sequence galaxies at z approximate to 1 have a high Hi mass, their short Hi depletion timescale is likely to cause quenching of their star formation activity in the absence of rapid gas accretion from the circumgalactic medium.