Is it possible to program an Arduino Pro Mini using an Arduino Uno R3?

Just hook up the pins (RX, TX, reset, Power, and Ground) to the corresponding pins on your Uno.

You may have to take the ATMega328 out of your Uno for this to work.

Keep in mind that you need to hook it up to the correct power pin! If your pro mini is running at 3.3v then you must hook it up to the 3.3 pin on your Uno.

Check out the ArduinoToBreadboard page for a bit more info. Scroll down to "Uploading Using an Arduino Board". Although it shows you how to program an Arduino on a breadboard, the wiring is the same.

you can program with an SMD board, you dont link the reset line, hold down the reset on the uno throughout the whole process, and tap the reset on the mini soon as the ide says upload to upload the sketch, by holding the uno in reset it doesn't interfere with the serial comms

There is another method by which you can upload your sketch programs. Program the UNO with the "Arduino as ISP" sketch and connect the pro mini via the SPI pins (plus reset line). See "Upload with Programmer" on the Arduino website and elsewhere on the internet including stack exchange. This method will allow you to use the full memory of the chip and avoid using the bootloader, but you will need to upload with SPI each time.