Soldering arduino I/O pins
You could do something like remove the headers of the pins on the arduino, solder in bigger male headers, and solder the other end of the male headers on whatever board you want. I've done it before on cheaper Arduino-like boards.
But unless you're fast and efficient at soldering, I suggest you stick to jumpers. Leaving the solder in contact with the Arduino too much is risky.
Another option is, as Passerby said, to use a bare ATMega328P, with the corresponding IC base instead.