Check the label on the side of the Dell Power supply. You are looking for the power rating in watts (not that important) and the current rating on the +12V, +5V & +3.3V outputs (very improtant). Post the numbers in the thread.
Also, see if the PSU label says "ATX standard" on it anywhere. That will tell you it is standard size. Dell has gotten better about non standard power supplies in the last few years.
Dell usually puts pretty robust power supplies in so you may not need an upgrade. On the other hand if any of that hardware was not bought and installed by Dell they will void your warantee, so getting a new PSU whose manufacturer will replace the PC is it gets fried may be a good idea.
EDIT
OMG - never mind I pulled the 8300 specs:
Output Wattage: 250 Watts
Input Voltage (auto-sensing): 90 to 135V at 50/60 Hz; or 180 to 265 V at 50/60 Hz
Heat Dissipation: 853 BTU/hour (fully loaded computer without monitor)
Backup Battery: 3.0 V CR2032 coin cell
The backview does look like a standard ATX power supply, not the mini.