Less than a month ago I was unemployed. I put out a bunch of applications and I landed on a local job that had me start right away. The pay was a lot less than I had anticipated but did it anyway because I needed money for bills and what not. Two weeks after employment, I got a job offer with another company in another part of the world. I want to go with this new company, but I want to break off cleanly from my current job. Also, once I accept the offer, I have about a month before I relocate, so keeping this job as long as I can is greatly preferred.

Should I give my current employer a chance to counter the offer as a common courtesy? And given that I want to keep working for this company until I leave, when would be the best time to quit, and how should I do so in the most professional way possible?

EDIT: I also want to mention that I am "temp-to-hire" at my current employer.