Funneling acoustic waves through largely mismatched channels is of fundamental importance to tailor and transmit sound for a variety of applications. In electromagnetics, zero-permittivity metamaterials have been used to enhance the coupling of energy in and out of ultranarrow channels, based on a phenomenon known as supercoupling. These metamaterial channels can support total transmission and complete phase uniformity, independent of the channel length, despite being geometrically mismatched with their input and output ports. In the field of acoustics, this phenomenon is challenging to achieve, since it requires zero-density metamaterials, typically realized with waveguides periodically loaded with membranes or resonators. Compared to electromagnetics, the additional challenge is due to the fact that conventional acoustic waveguides do not support a cut-off for the dominant mode of propagation, and therefore zero-index can be achieved only based on a collective resonance of the loading elements. Here we propose and experimentally realize acoustic supercoupling in a dual regime, using a compressibility-near-zero acoustic channel. Rather than engineering the channel with subwavelength inclusions, we operate at the cut-off of a higher-order acoustic mode, demonstrating the realization and efficient excitation of a zero-compressibility waveguide with effective soft boundaries. We experimentally verify strong transmission through a largely mismatched channel and uniform phase distribution, independent of the channel length. Our results open interesting pathways towards the realization of extreme acoustic parameters and their implementation in relevant applications, such as ultrasound imaging, acoustic transduction and sensing, nondestructive evaluation, and sound communications.