I have the SDcard breakput board connected to promini like this:
Arduino-Sdcardbreakout board
3V- 3V
GND-GND
13-CLK
12-D0
11-D1
10-CS
The constant resets are a symptom of a power problem.
There are 3v and 5v issues to sort out. Do you have a 3.3v pro mini or a 5v promini?
What are you conecting 3v - 3v? Where are you getting the 3v from? I'm not sure you can take the 3V output from a 5V promini and get enough supply current.
As I understand the adafruit SD module, it usually takes a 5v input to an onboard regulator.
The power draw when writing to the card can be fairly high, up to 100mA (or more)! That means that you must have a fairly good 3.3V power supply for the card.
Please explain more about your wiring.