The Arduino Uno (for some models) has the ability to dislodge its IC chip. You can mount this on a piece of perfboard and, in essence, have a permanent connection. I will need the beefier Arduino Due, but I do not know how to make permanent connections with this device. Is there any way to form some kind of permanent connection with this board?
What it sounds like you want is an Arduino Due footprint prototyping shield.
Arduino make one, although it is out of stock, but it gives you an idea of what you are looking for. There are plenty of other designs out there for purchase of varying degrees of complexity.
You solder the pins in place and mount it on top of your Due just like any shield, and solder your wires and components in where you want them.
An alternative option, of course, is to step away from Arduino and investigate some of the many other chips from other manufacturers out there that are considerably more powerful than the 8-bit ATMegas used on the low-end Arduinos, and yet are still available in a DIP footprint - chips like the PIC32MX1xx/2xx series from Microchip - 28 pin DIP (just like the 328P) yet 50MHz, 32-bit power. Yes, it will be a bit of a learning curve, but there are helpful options out there that can assist you, like the chipKIT system including the handy DP32 board from Digilent.
I think the best thing you can do with this, is that you use temporary wires to connect and test the application and later, when you'll come to part, where you should "make-a-solid", you just solder these wires to the bottom, where pin headers are. I think this is the simplest way to do it.
If you want to do it a bit better, desolder female header pins and solder cables to those holes. It's more electronics-approved that way.
Or the simplest (but not the brighest): Jumper wires can be glued into female header pins, but I strongly unrecommend this option, as same as making a shield for this.
There's another(and most elegant too!) option of making a PCB version of arduino: Add the AtMega, few passive components and proper pads, to solder wires on.