End the last round with your yarn.
After completing the round, slip the last stitch back on to the left hand needle.
Switch to your new yarn and knit this last stitch again with it.
Continue to work the next round with your new yarn, until you've reached the last stitch before the Beginning of the Round.
Go into the stitch below from the previous round and ...
... lift it on to the left hand needle.
Knit both stitches (old yarn & new yarn) together as one with your new yarn.
Readjust the yarn ends a little to align the stitches and continue working with your new yarn.
Seamless Yarn Transition
If you are knitting a striped piece, you may be annoyed by the small steps that are clearly visible at the beginning of a round when you change yarn.
So I'm going to show you a little trick that you can use to make this transition a little nicer and seamless. This can make this step almost completely invisible.