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The development of high-current cables made of rare-earth barium copper oxide (REBCO) coated conductors is a key technology for high-field accelerator magnets above 16 T. A flexible round REBCO cable has been proposed as one of the candidates for the high-current cable. This round REBCO cable consists of many coated conductors wound in multiple layers in a helical configuration on a metal core. The current transport properties of the round REBCO cable at 4.2 K and in high magnetic fields are essential for the design of high-field accelerator magnets. In this study, a solenoid-wound bobbin made of glass fiber reinforced plastic (GFRP) was developed to characterize the critical current (Ic) of the round REBCO cable at 4.2 K and 18 T. A 2-mm-wide coated conductor, manufactured by SuperPower Inc., was used to fabricate the sample cable. The Ic of the coated conductor was also measured at 4.2 K for comparison with that of the sample cable. The Ic and lift factors (Ic(4.2 K)/Ic(77 K)) of the coated conductor and the sample cable were reported, and the estimation of the cable Ic using the conductor Ic was discussed. As expected, the lift factors of the coated conductor and the sample cable were almost the same after correcting for the self-field effect.