There are no DDR5 adapters for DDR4 motherboards.
Yes, you need a new motherboard — and likely a new CPU — to use DDR5 RAM. No adapter exists for desktop memory.
DDR4 and DDR5 use different physical slots, voltage regulators, and signaling. The motherboard’s memory controller is built into the CPU, and that controller is wired for one generation only. Intel’s 12th, 13th, and 14th gen support both, but only on boards designed for DDR5. AMD’s Ryzen 7000 and later are DDR5-only.
Even if someone somehow made an adapter, the electrical differences would make it unstable or slow. Nobody makes them for consumers because it’s not technically feasible without major redesign.
If you’re upgrading to DDR5, you’re buying a new motherboard and probably a new CPU. Plan for that cost.
