In just about 4 years, IPv6 will celebrate its 20th anniversary. While the protocol itself is already quite old, its deployment has only recently picked up speed. Not so many Internet service providers offer direct IPv6 connectivity to their customers, yet. Clearly, when IPv6 is available to customers, they expect that IPv6 offers at least the same - or even better - stability of connections in comparison to IPv4. The main goal of this paper is to investigate whether this is true today. In our paper, we present up-to-date measurement results on the stability of IPv4 and IPv6 paths in the real Internet, based on machines that are distributed over a large geographical area, as part of the NorNet Core testbed infrastructure for multi-homed systems. The measurements not only cover high-speed research networks, but also consumer-grade ADSL connections - i.e. the ISP connection types of "normal" end-users - as well as a broad range of different ISPs. The measurements show that IPv6 paths are less stable than corresponding IPv4 paths. We also find that the use of load balancing is more prevalent in IPv6 than in IPv4.