Depending on the sound card, this may be a classic hum loop issue... Some sound cards "pretend" to be real balanced inputs, like the M-Audio cards, but they are not really. If you're depending on a pre-made cable, you'll be in trouble. If you're not sure the card is a real balanced input, make a cable with a mono 1/4 plug to go into the sound card, that goes to an XLR on the codec, with Pin 1 on the XLR going to shield, and pin 3 going to the center conductor (yes, I know this is unbalanced)... if you have other devices that this codec output goes to, make sure you wire them carefully. If you want to bridge a balanced input device across the connections on the output of the codec, you should have no trouble, as the (+) and (-) inputs to the balanced device (a console input, for example), never get connected to shield at any point in the wiring.
Remember, it is easy to get a balanced input to accept an unbalanced output, and it is easy,(at least with modern equipment) to drive an unbalanced input from a balanced device, but care has to be given to the wiring.
ALWAYS wire an input, regardless if it is a transformer or transformer-less coupled input as if it IS a transformer coupled input, meaning, no input should EVER be left floating. If it is not signal, connect it to the shield. At the output-end of an input wire, it is usually good practice to just connect the (+) and (-) without connecting the shield.
On outputs, the opposite is true, if you want to use an active balanced output as an unbalanced output, connect to shield and the positive output only. Don't ever connect an + or - pin to shield as these usually have signal on them that will get shorted to ground, usually with bad results.
In your situation, the problem arises when the ground is connected at one end, and goes into one of the balanced input pins on the other end (or likewise is connected to the (+) and (-) at the output), without also being connected to shield. This sets up the "hum loop" because the balanced input gets it's signal path through another source, say the AC lines between the equipment. A pre-made cable, can potentially do this, and if other combinations of equipment are connected to the same output, they can produce similar results, because even if both the sound card and the codec are balanced, maybe someone else made a wiring error that caused shield to get connected to the (-) wire, for example? If your signal comes from the codec and shield is not connected and you're expecting to get an unbalanced tap somewhere at the studio, you may need to connect the shield at the codec.
One time I saw an excellent tutorial in an old Arrakis console manual on how to combine different types of equipment, and the method works flawlessly. I wish I had the illustration for every engineer, because it will save you hours of headaches just like this. What I said above is a verbal description of what I saw in that illustration. I hope it makes sense to you.
If you cannot solder your own connectors onto wire, I'd suggest finding someone to help you, as this is a VERY fixable situation... you should be able to create a noise-free signal from the codec.