A sub-nanometer trans-electrode membrane made of graphene has been demonstrated. Graphene, an isolated, atomically thin conducting membrane of graphite, exhibits new electrochemical properties when immersed in ionic solution, making it a trans-electrode. The trans-electrode properties arise from the atomic-scale proximity of its two opposing liquid-solid interfaces and graphene's in-plane conductivity. A CVD-grown graphene membrane, only one to two atomic layers thick, was found to be a remarkable ionic insulator with very small stable conductivity, dependent on the ion species in solution. Electrical measurements on graphene membranes with a single nanopore showed that the membrane's effective insulating thickness is less than one nanometer, making graphene ideal for high-resolution, high-throughput nanopore-based single molecule detectors. The trans-electrode ionic conductivity of a graphene membrane was measured by mounting a CVD-grown sheet of graphene over an aperture in a SiNₓ layer. The membrane was inserted into a fluidic cell, separating two compartments filled with ionic solutions. With a 100 mV bias applied between Ag/AgCl electrodes, ionic current measurements showed the graphene membrane’s trans-ionic conductance was far below the nS level. The highest conductivities were observed for solutions with the largest atomic size cations, likely due to their minimal hydration shell. Drilling a single nanometer scale pore in the graphene trans-electrode membrane increased its conductivity by orders of magnitude, enabling experiments to evaluate the graphene electrode’s effective insulating thickness. The ionic conductivity G from a pore of diameter d in an infinitely thin insulating membrane is given by G_thin = σ·d, where σ is the conductivity of the ionic solution. The current density in the case of a nanopore in a very thin membrane is sharply peaked at the pore’s perimeter. For finite but small thicknesses, computer simulations were used to predict the conductivities. Measurements of nanopore conductivity while it is being traversed by an insulating long chain polymer of DNA provided an alternative method of evaluating the graphene insulating thickness. The results using a 5 nm pore in graphene and double stranded DNA molecules showed that the membrane thickness and nanopore diameter can be determined. The best fit to the measured pore conductance data yielded L_GIT = 0.6 nm, in excellent agreement with the value deduced from open pore measurements. The pore diameter d_GIT = 4.6 nm also agreed with the geometric diameter of 5 nm obtained from TEM. The extremely small L_GIT value suggests that nanopores in graphene membranes are uniquely optimal for discerning spatial or chemical molecular structure along the length of a molecule as it passes through the pore. The model assumes a long insulating 2.2 nm diameter cylinder symmetrically translocating through the center of a 2.4 nm diameter nanopore. Solving the conductivity for this geometry as the discontinA sub-nanometer trans-electrode membrane made of graphene has been demonstrated. Graphene, an isolated, atomically thin conducting membrane of graphite, exhibits new electrochemical properties when immersed in ionic solution, making it a trans-electrode. The trans-electrode properties arise from the atomic-scale proximity of its two opposing liquid-solid interfaces and graphene's in-plane conductivity. A CVD-grown graphene membrane, only one to two atomic layers thick, was found to be a remarkable ionic insulator with very small stable conductivity, dependent on the ion species in solution. Electrical measurements on graphene membranes with a single nanopore showed that the membrane's effective insulating thickness is less than one nanometer, making graphene ideal for high-resolution, high-throughput nanopore-based single molecule detectors. The trans-electrode ionic conductivity of a graphene membrane was measured by mounting a CVD-grown sheet of graphene over an aperture in a SiNₓ layer. The membrane was inserted into a fluidic cell, separating two compartments filled with ionic solutions. With a 100 mV bias applied between Ag/AgCl electrodes, ionic current measurements showed the graphene membrane’s trans-ionic conductance was far below the nS level. The highest conductivities were observed for solutions with the largest atomic size cations, likely due to their minimal hydration shell. Drilling a single nanometer scale pore in the graphene trans-electrode membrane increased its conductivity by orders of magnitude, enabling experiments to evaluate the graphene electrode’s effective insulating thickness. The ionic conductivity G from a pore of diameter d in an infinitely thin insulating membrane is given by G_thin = σ·d, where σ is the conductivity of the ionic solution. The current density in the case of a nanopore in a very thin membrane is sharply peaked at the pore’s perimeter. For finite but small thicknesses, computer simulations were used to predict the conductivities. Measurements of nanopore conductivity while it is being traversed by an insulating long chain polymer of DNA provided an alternative method of evaluating the graphene insulating thickness. The results using a 5 nm pore in graphene and double stranded DNA molecules showed that the membrane thickness and nanopore diameter can be determined. The best fit to the measured pore conductance data yielded L_GIT = 0.6 nm, in excellent agreement with the value deduced from open pore measurements. The pore diameter d_GIT = 4.6 nm also agreed with the geometric diameter of 5 nm obtained from TEM. The extremely small L_GIT value suggests that nanopores in graphene membranes are uniquely optimal for discerning spatial or chemical molecular structure along the length of a molecule as it passes through the pore. The model assumes a long insulating 2.2 nm diameter cylinder symmetrically translocating through the center of a 2.4 nm diameter nanopore. Solving the conductivity for this geometry as the discontin