This paper presents a stochastic method for calculating the ground state of an electron gas, also known as the Fermi one-component plasma or jellium. The method uses a Monte Carlo simulation to estimate the correlation energy and determine phase transitions at zero temperature. The study shows that at intermediate densities, a ferromagnetic electron fluid is more stable than an unpolarized state. The method involves a two-step process: first, a fixed-node approximation is used to calculate the energy, which serves as an upper bound to the exact energy. Second, nodal relaxation is performed to refine the energy estimate by allowing random walks to cross nodes and reversing their contribution to averages. The method is based on a stochastic simulation of the Schrödinger equation for charged bosons and fermions. The trial wavefunction is a product of two-body correlation factors and a Slater determinant of single-particle orbitals. The simulation uses a variational Monte Carlo approach and involves a trial wavefunction of the Bijl-Jastrow-Slater type. The results show that the energy of the electron gas is highly dependent on the density parameter $ r_s $, which is the Wigner sphere radius in units of Bohr radii. The study finds that the Fermi system undergoes two phase transitions: crystallization at $ r_s = 100 \pm 20 $ and depolarization at $ r_s = 75 \pm 5 $. The energy difference between a Bose crystal and a Fermi crystal is less than $ 1.0 \times 10^{-6} R $ at $ r_s = 10^0 $. The results are compared with other theoretical methods, showing that the coupled-cluster formalism gives the most accurate results. The study also highlights the importance of exact simulations in reliably calculating phase transitions and densities. The authors thank M. H. Kalos for his contributions and acknowledge the computational assistance of Mary Ann Mansigh. The paper concludes that the method provides a reliable way to calculate the ground state energy of the electron gas with high accuracy.This paper presents a stochastic method for calculating the ground state of an electron gas, also known as the Fermi one-component plasma or jellium. The method uses a Monte Carlo simulation to estimate the correlation energy and determine phase transitions at zero temperature. The study shows that at intermediate densities, a ferromagnetic electron fluid is more stable than an unpolarized state. The method involves a two-step process: first, a fixed-node approximation is used to calculate the energy, which serves as an upper bound to the exact energy. Second, nodal relaxation is performed to refine the energy estimate by allowing random walks to cross nodes and reversing their contribution to averages. The method is based on a stochastic simulation of the Schrödinger equation for charged bosons and fermions. The trial wavefunction is a product of two-body correlation factors and a Slater determinant of single-particle orbitals. The simulation uses a variational Monte Carlo approach and involves a trial wavefunction of the Bijl-Jastrow-Slater type. The results show that the energy of the electron gas is highly dependent on the density parameter $ r_s $, which is the Wigner sphere radius in units of Bohr radii. The study finds that the Fermi system undergoes two phase transitions: crystallization at $ r_s = 100 \pm 20 $ and depolarization at $ r_s = 75 \pm 5 $. The energy difference between a Bose crystal and a Fermi crystal is less than $ 1.0 \times 10^{-6} R $ at $ r_s = 10^0 $. The results are compared with other theoretical methods, showing that the coupled-cluster formalism gives the most accurate results. The study also highlights the importance of exact simulations in reliably calculating phase transitions and densities. The authors thank M. H. Kalos for his contributions and acknowledge the computational assistance of Mary Ann Mansigh. The paper concludes that the method provides a reliable way to calculate the ground state energy of the electron gas with high accuracy.