Yes, you can do it without a problem. Just solder each wire carefully and use fine shrink tubing over it and be sure to wrap the shielding where it's necessary. Seal the whole joint area with glue lined shrink tube or cover it with evostick and shrink normal shrink tube. If you do the latter, heat from the middle of the joint outward to expel any air and solvent vapour.
When you've opened the cable, shorten the cores a little but leave the shielding long so you have a bit to play with when rewrapping.