It looks like black tip. You need a fungicide rather than an insecticide. I believe black tip can live over the winter on your plant. Perhaps you need to keep treating with a fungicide over the course of winter so that it does not spread.

Also, I think you should be watering only on the soil and not letting the leaves get wet.

Hopefully someone with black tip experience will come along soon and give you some better advice.

Just judging from that picture, the stalk looks a bit wrinkled also. Watering once a week may be too much, could possibly have caused root rot. Plumeria's like to be on the dry side, a bit like cactus.

Black tip can be caused by stress also, perhaps from too much water.

Others with more experience with this can give advice at what to do at this stage.

Is it a newly-rooted cutting? Was it rooted when you got it? Please share with us some of the history of this plant.

It's a blurry photo, but it does look like the stem is wrinkly and that says to me the root system is compromised...either from too much water or it's just not had time to get established yet.

I would hold back on water, keep it in semi-shade, don't jiggle the plant a lot (breaking roots) and hope for the best.

Black tip is not a fungus, so fungicide will do nothing.

Keep an eye on the tip. Usually the necrosis will stop and it will just dry up. IF the stem feels squishy, there could be more damage than we're seeing.

I think the replies about too much water maybe correct, although I didn't think I was giving it that much maybe 1/4 cup once a week. When I first started the cutting I didn't water at all and it was doing great but I read somewhere once it is rooted I should water very lightly. I am going to stop watering for now and hope it comes back. Should I cut the black stuff off?
Thanks for all the replies

For anyone interested or confused about Dave saying black tip is not fungal, check this out from an older post....

"Black tip in NOT caused by fungal issues according to the latest research. Several of us on the West Coast have sent samples of "black tip" to labs. The conclusion of the labs testing on freshly harvested new appearances of stems with "black tip" was "no fungus found". With older stems there were findings of fungal (fuserium being one) activity leading to the conclusion that fungal activity was secondary. Another study by Sean Miller and recognized by Dr. Criley was; mites on the tip of stems cause stress to the plant and the plumeria develops "black tip"."

See this link:
http://plumeriatc.org/ - some interesting info here.