A Fabry-Perot cavity polarimeter, installed in 2003 at HERA for the second phase of its operation, is described. The cavity polarimeter was designed to measure the longitudinal polarisation of the HERA electron beam with high precision for each electron bunch spaced with a time interval of 96 ns. Within the cavity the laser intensity was routinely enhanced up to a few kW from its original value of 0.7 W in a stable and controllable way. By interacting such a high intensity laser beam with the HERA electron beam it is possible to measure its polarisation with a relative statistical precision of 2% per bunch per minute. Detailed systematic studies have also been performed resulting in a systematic uncertainty of 1%.To cope with the physics program after the upgrade, a fast and high precision longitudinal Compton polarimeter using a continuous wave laser resonating in a Fabry-Perot cavity (LPOL cavity) was proposed [4], constructed and installed near the existing longitudinal Compton polarimeter (LPOL) [5]. In addition to the LPOL, the transverse polarisation of the electron beam is also measured by another polarimeter (TPOL) [6].With respect to the prior HERA LPOL and TPOL polarimeters, the higher statistical precision of the LPOL cavity is achieved by increasing firstly the power of the continuous wave laser by more than two orders of magnitude compared to the TPOL and secondly the frequency of the electron-photon(laser) interaction to 10 MHz compared to 0.1 kHZ of the pulsed laser of the LPOL. A new Data Acquisition System (DAQ), synchronised to the HERA beam clock, has been developed accordingly which operates without any trigger at 10 MHz. This is one of the novelties of the experiment described in this article.The HERA Fabry-Perot cavity is similar to a device that has been used successfully to measure the polarisation of the CEBAF LINAC electron beam [7,8,9]. One major difference between the HERA and CEBAF LPOL cavities is the dynamical regime. Whereas the luminosity of Compton scattering is relatively low at CEBAF, it reaches much higher values at HERA. That is, the average number of scattered Compton photons is close to one per bunch in the latter case and much smaller in the former. The HERA dynamical regime, denoted as 'few photon mode' in this article, has been used successfully for the first time by the LPOL cavity to measure the electron beam polarisation. An important point to mention is that a huge effort was made to reduce the unforeseen high level of synchrotron radiation emitted by the electron beam at the cavity location downstream of the HERMES experiment, by adding many protections (cf Sect. 3.1 for more detail) to avoid damaging any optical or electronic component [10]. Especially fragile were the 'supermirrors', whose high quality, which was maintained until the end of data taking, was the key to obtaining the foreseen high power of the laser,The main purpose of this article is to describe the LPOL cavity experiment and to report about the electron polarisation measurement. The article is org...