We formulate a quantum master equation for the many-particle density matrix of electrons propagating through a single-mode conductor, combining elastic scattering by disorder with time-resolved projective measurements that monitor the outcome of scattering events. For noninteracting electrons the charge transfer statistics is governed entirely by the single-particle sector of Fock space, which allows us to resum the exponentially many measurement outcomes into a master equation of dephasing-channel form. The full counting statistics of transmitted electrons has a binomial distribution function, whose mean T and variance T (1 −T ) determine the conductance and shot noise power, respectively. Monitoring suppresses the phase coherence responsible for one-dimensional localization: The decay with conductor length L of the typical transmission probability crosses over at L ≃ ℓϕ from the exponential e^{−L/ξ} (with localization length ξ) to the Ohmic 1/L decay. Numerical solution of the master equation gives, for weak monitoring, a logarithmic dependence ℓϕ ≃ ξ ln(vFτϕ/ξ) of the coherence length ℓϕ on the mean time τϕ between measurements.