What do you do when you finish a project?
There are many options, so it all ends up depending on your personal
preferences:

You could indeed dedicate this Arduino to this project, and buy a new
one for the next project. This may be the simplest solution, if not
the cheapest. Just make sure you are not using a breadboard for the
other parts. A “proto-shield” may be a good option for soldering the
rest of your circuit in a permanent form.

You may replace the Uno with a smaller Arduino, like the Micro (but
beware it uses a different MCU) or a Pro Mini clone (same MCU). These
smaller boards can then be soldered onto a PCB using standard 0.1” pin
headers.

You could buy a cheap clone, test it to make sure it works as intended
on this particular project, and dedicate the real one for further
prototyping.

You could get a bare ATmega328P plus crystal and caps and put them,
together with the rest of your circuit, onto a perf-board or a
strip-board. C.f. Michel Keijzers’ answer. Note that if your program
can run happily off a somewhat inaccurate 8 MHz clock, you can
use the MCU's internal oscillator instead of a crystal.

If your program is far from using all the resources of the Uno, you
may consider a smaller MCU. The ATtiny85 is a popular “minimalistic”
choice. Make sure an Arduino core is available for it, unless you want
to program all low-level.

For the most professional result, you could design and order a real
PCB for your whole circuit, including the MCU.