No, the Neanderthal genome has not been completed

The new genome (in red) is shown in relationship to other human genomes.

http://www.eva.mpg.de/neandertal/index.html

Today, the people behind the Neanderthal genome project (Svante Pääbo's group at the Max Planck Institute) loosed yet another Neanderthal genome on the world, a genome that many press outlets are calling "complete." This one comes from a single toe bone found in the same cave as the finger bone that helped us identify the Denisovans, an extinct group of humans who inhabited Asia tens of thousands of years ago. Apparently, something about the cave environment has preserved ancient DNA extremely well.

That good preservation, combined with improved techniques, has provided a very high quality new sequence. Genome sequences are obtained randomly, typically in stretches about a hundred bases long. Over something the size of a human genome (about 3 billion bases), this randomness means that some areas will be sequenced many, many times, and others will be missed entirely or contain errors that repeated sequencing would allow us to catch. The typical measure of quality for a genome is called "coverage," and that's simply the number of times the average base would be sequenced if every base was covered equally.

In this case, the genome has 50-fold coverage. That's well above draft quality (which tends to be around 30-fold), but there will definitely still be gaps, errors, and places in the genome that aren't sequenced at all. Still, that's about as good as we're likely to get with ancient genomes; the sequence includes almost all of the non-repetitive DNA found in the human genome, and it will provide a valuable resource for comparative studies.