A remarkably high number of people have suggested that James Watson was referring to a field that is outside his field of expertise - or even unrelated to DNA. I don't know how many of those people actually believe that race is not encoded in DNA but the number of people who have verbally contradicted basics of elementary school biology in this fashion is incredibly high anyway. Color of plants and color of other things are the first examples that are discussed whenever genetics is taught at school.
The skin color gene
Let me be very specific. The relevant gene is called SLC24A5. The acronym means that it is included in the solute carrier family and it is the member 5. The gene has 21,420 base pairs (plus minus one because of two possible deletions). See the sequence. If you care where it is, it is on the long (q) arm of chromosome 15, on position 21.1, starting from base pair 46,200,461 and ending on 46,221,880 (no Java).
The gene is expressed in brain (!), hindbrain(!), medulla oblongata (!), nervous system (!), and cone photoreceptors.
Now, look at the 111th amino acid. You will find that in between 93 and 100 percent of the East Asian, native American, and African population, the 111th amino acid is alanine (A which is encoded as GCU, GCC, GCA, or GCG): the allele (version) of the gene is then called ala-111 for obvious reasons. In something like 98.7 - 100 percent of the European population, the 111th amino acid is threonine (T which is encoded as ACU or ACA) and the allele is then called thr-111. See the sequence (GCA or ACA) of Craig Venter's personal DNA code around the ambiguous nucleobasis (G/A) number 15 46213776. It is one of only six nucleobases in the 22,420-base-pair-long gene where variations are known.
You shouldn't confuse alanine and threonine (amino acids, building blocs of proteins) with (thrice) simpler blocs of DNA, the small letters or nucleobases, namely adenine, thymine (or its counterpart uracil in RNA), cytosine, and guanine.
It is estimated that this allele - a single bit that is changed and that also influences pigmentation - only appeared 6,000-10,000 years ago or so. The Biblical Adam could have been the first white person, after all... These insights have only been known since 2005 (see e.g. the Washington Post).
Because the white genes are mutations of the genes of the original men of color - and males are mutations of the original females - we can finally answer the question "Is God black?" The answer is "Yes, She is." Entertainingly enough, the gene also exists and decides about the color in other species. It has been demonstrated in zebrafish, a common research proxy for evolution biology. ;-)
Figure 1: Races of zebrafish. To be sure: the left fish is African American while the right fish is Caucasian. As far as I am concerned, the original N-word zebrafish is prettier but I don't know which of them is smarter.
Knowledge can't be undone
There are surely several additional genes, for example SLC45A2, that carry the information about race - and distinguish the remaining races, among other things - but this one is arguably the key. Although it is just one base among more than 3 billion bases, it accounts for at least 1/3 of the differences in skin color. Some people like to say that everything is multi-dimensional and complicated but certain things are really stored in one or several base pairs. They are not complicated at all once they are understood.
I wonder what the people who find the biological origin of races (and perhaps even sexes?) to be a taboo want to do with all this scientific knowledge. They would not only have to erase all the 22,000+ web pages and burn the books that contain this information but also execute everyone who knows what is the answer and suppress the curiosity of all new students of biology as well.
I am afraid that this won't and can't work. The genetic origin of many features of organisms, including humans, starts to be understood and it is all but inevitable that the degree of our understanding of these issues will keep on increasing. For example, you can download Craig Venter's and James Watson's genomes from this FTP website. If someone wants to build the future upon the assumption that these things will remain unknown, he or she is preparing the soil for some very bad conflicts and he or she will surely stay on the wrong side.
And that's the memo.