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To achieve high temporal resolution for ultrafast electron diffraction, Zewail (Proc. Natl Acad. Sci. USA 102, 7069 (2005)) has proposed to use high repetition rate, ultrafast electron sources. Such electron sources emitting one electron per pulse eliminate Coulomb broadening. High repetition rates are necessary to achieve reasonable data acquisition times. We report laser-induced emission from a nanometre-sized tip at one electron per pulse with a 1 kHz repetition rate in the femtosecond regime. This source, combined with 1 MHz repetition rate lasers that are becoming available, will be a primary candidate for next generation ultrafast, high-coherence electron diffraction experiments. We also report that the measured energy bandwidth of our electron source does not support sub-cycle electron emission. This result addresses a current debate on ultrafast nanotip sources. Regardless of the limited bandwidth, this source may be used in conjunction with a recently proposed active dispersion compensation technique (Proc. Natl Acad. Sci. USA 104, 18409 (2007)) to deliver attosecond electron pulses on a target.