Thread as skinny a needle as you can use with your yarn and go back into the strand of yarn about 3" back from end. Try to run the yarn inside the plies if possible. It helps to untwist the plies slightly to open up the center.
Pull the needle all the way thru the yarn leaving a loop at the end. Take the 2nd yarn end (pink) and pass thru this loop.
Repeat Step 1 with this 2nd piece of yarn so you have 2 interlocked loops with each end threading back thru its own strand.
Pull the free ends down so the loops disappear and trim the ends off where they exit the strand (see where needle is pointing). Pull firmly to make sure they are long enough to hold. When you close down the loops in Step 4, just pull until closed. If you keep pulling, the strand will untwist and release the yarn end in the center of the strand.
The finished join. This one is a little short. I like to have at least 2" of "yarn thru yarn" on each side of the joining.