This work presents the experimental results and analytical modelling of the cold end of a closed-cycle gravity-insensitive dilution refrigerator adapted from the open-cycle dilution refrigerator used for the Planck mission. The refrigerator is designed to provide 1 μW of cooling at a temperature of 50 mK. The cold end of the refrigerator comprises a counterflow heat exchanger (which pre-cools the 3He and 4He components down from a temperature of about 1 K to below 100 mK), a mixing chamber and a load heat exchanger at about 50 mK. We discuss the counterflow heat exchanger designs that were considered and present experimental results. The best configuration shows a cooling power of 1 μW at 45 mK. We develop analytical models for the heat exchangers and the mixing chamber and compare them with experimental data.