Oxy-fuel combustion is one of the main routes for carbon dioxide (CO2) capture in power plants. The technology is well-developed for coal-fired power plants but is less explored for natural-gas-fired gas turbine cycles. Implementing oxy-fuel CO2 capture in gas turbines is more complex than in boilers because the power density is larger and the working fluid of the power cycle is changed from air to CO2. The combustion system must then handle the combustion of the fuel with pure oxygen (O-2) and the recirculated exhaust gas composed of mainly CO2. In this study, the pressurized combustion of methane in O-2/CO2 atmospheres in a pressurized oxy-fuel combustion facility (HIPROX) is presented. The experiments focused on flame stability and CO emissions, which are potentially challenging in this type of combustion. The experimental setup is based on an in-house axial swirl-stabilized burner placed inside an optical combustion chamber made of quartz. Flame stability and CO emissions were studied at different O-2 concentrations, excess O-2 ratios, fuel power loads up to 100 kW, and pressures up to 10 bar. Concerning stability, the trade-off between stability and excessive temperature was clearly evidenced and quantified for that burner. Dependent upon pressure and power loads, an O-2 concentration of about 30% resulted in a stable safe flame for most conditions, while blow-off could occur at an O-2 concentration from 23 to 29% depending upon power and pressure. The experimental results show that special considerations have to be taken with respect to the CO formation when implementing oxy-fuel combustion in gas turbine conditions. High equilibrium CO concentrations at flame temperature combined with short residence times at high and intermediate temperatures can lead to high emissions of CO. The CO emissions were found to be highly dependent upon excess O-2, and although there is a strong decreasing trend with increasing O-2 excess, even with 10% O-2 excess, the CO values were excessively high. It was found that the CO emissions are partly controlled by the equilibrium CO concentration and, therefore, increased with an increasing O-2 concentration in the oxidizer as a result of an increasing adiabatic flame temperature.