  This report presents a project of an experiment for precision studies of the
elastic electron proton scattering in the low momentum transfer region. The
project is based on an innovative experimental method which allows for
detection of recoil protons in a hydrogen "active target". The goal of the
experiment is to measure the $ep$ elastic scattering differential cross section
in the $Q^2$ range from 0.001 GeV$^2$ to 0.04 GeV$^2$ with 0.1% relative and
0.2% absolute precision and to determine the proton charge radius with a
sub-percent precision. The important advantages of the proposed experiment are
low radiative corrections, inherent to the recoil proton method, and absolute
measurements of the differential cross sections. The experiment will be
performed in the 720 MeV electron beam of the Mainz electron accelerator MAMI.
This accelerator can provide an electron beam with optimal for this experiment
parameters. The Proposal was approved by the MAMI Program Advisory Committee,
and a special Agreement aimed at realization of this experiment was signed
between Petersburg Nuclear Physics Institute and Institute of Nuclear Physics
of the Mainz University. Also, scientists from the following institutes will
participate in this project: GSI Centre for Heavy Ion Research, Germany; Joint
Institute for Nuclear Research, Russia; William and Mary College, USA; Mount
Allison University, Canada; University of Regina, Canada; Saint Mary's
University, Canada. The first measurements should be started in 2020.
