New results from a large survey of Hα emission-line galaxies at z = 0.84 using the Wide Field Camera on the United Kingdom Infrared Telescope and a custom narrow-band filter in the J band are presented as part of the High-z Emission Line Survey (HiZELS). The deep narrow-band images reach an effective flux limit of F _Hα ∼ 10 ^-16 erg s ^-1 cm ^-2 in a comoving volume of 1.8 × 10 ⁵ Mpc ³, resulting in the largest and deepest survey of its kind ever done at z ∼ 1. There are 1517 potential line emitters detected across ∼1.4 deg ², of which 743 are selected as Hα emitters, based on their photometric and spectroscopic redshifts. These are then used to calculate the Hα luminosity function, which is well fitted by a Schechter function with L* = 10 ^42.26±0.05 erg s ^-1, φ* = 10 ^-1.92±0.10 Mpc ^-3 and α = -1.65 ± 0.15, and are used to estimate the volume average star formation rate (SFR) at z = 0.845, ρ _SFR: 0.15 ± 0.01 M _⊙ yr ^-1 Mpc ^-3 (corrected for 15 per cent active galactic nucleus contamination and integrated down to 2.5 M _⊙ yr ^-1). These results robustly confirm a strong evolution of ρ _SFR from the present day out to z ∼ 1 and then flattening to z ∼ 2 using a single star formation indicator: Hα luminosity. Out to z ∼ 1, both the characteristic luminosity and space density of the Hα emitters increase significantly; at higher redshifts, L* continues to increase, but φ* decreases. The z = 0.84 Hα emitters are mostly disc galaxies (82 ± 3 per cent), while 28 ± 4 per cent of the sample shows signs of merger activity; mergers account for ∼20 per cent of the total integrated ρ _SFR at this redshift. Irregulars and mergers dominate the Hα luminosity function above L*, while discs are dominant at fainter luminosities. These results demonstrate that it is the evolution of 'normal' disc galaxies that drives the strong increase in the SFR density from the current epoch to z ∼ 1, although the continued strong evolution of L* beyond z = 1 suggests an increasing importance of merger activity at higher redshifts.