Then he very carefully defines, in poetic terms, the worlds within and without. He says that he "leaves the white world" and that he steps "within the Veil,"which is theantithesis, or the world of black folk. He goes on to say that the reason he does this is so that the "Gentle Reader," the white reader, might see the "deeper recesses" within the Veil: "its religion, the passion of its human sorrow, and the struggle of its greater souls." In this description, "its" represents a metonymy in which Veil now stands for the lives of black folk who dwell their whole lives within the Veil. To make matters even more certain, Du Bois adds at the end of "The Forethought" a most poetic and poignant self-identification that positions his own life precisely within the Veil:

need I add that I who speak here am bone of the bone and flesh of the flesh of them that live within the Veil?

Yet Du Bois' use of this symbol is far more complex than that, making it all that much more confusing to understand if you are not alert in following his poetic prose. In Chapter 1, he defines the veil itself by describing his first instance of recognizing the divide that separates him and his folk from white folk with "stringy heads." He describes that "it dawned upon me with a certain suddenness that I was different from the others." He explains that he was "shut out from their world by a vast veil." He also says he felt contempt for those white folk on the other side of the veil. So we now we have two meanings to veil. The first is "the Veil" that contains the souls of black folks "within the Veil" and the second is "the veil" that separates black folk, who are like white folk "in heart and life and longing," and leaves them "shut out from their world by a vast veil."

There are two other uses for "veil" in the first three chapters. One is the idiomatic expression "born with a veil," which refers to a condition at birth in which a thin membrane veil covers the face of the new born. In Du Bois’ allusion to it, this veil refers to an ability in psychic powers. We can confirm this because "born with a veil" is used in conjunction with "seventh son," another idiomatic expression that may mean a son born with special powers. Du Bois links these special powers to psychic abilities with yet another idiomatic expression, "second sight," which refers to clairvoyance, a psychic ability that Du Bois has thus attributed to black folk within the Veil..

The third use for "veil" is in Chapter 2. Du Bois constructs a powerful metaphor comparing a land "right merry with the sun" to a "figure veiled and bowed" in the "King's Highways." America of white folks is the land where "children sing" and the unembraced black folks of the "color-line" is the figure draped in a veil of shame or mourning sitting in the midst of plenty, bowed down and with nothing but the veil.

I have seen a land right merry with the sun, where children sing, and rolling hills lie like passioned women wanton with harvest. And there in the King's Highways sat and sits a figure veiled and bowed, by which the traveller's footsteps hasten as they go. ... the problem of the color-line.

In The Souls of Black Folk, W.E.B. DuBois repeatedly refers to this concept of "the veil" to describe the separation between black Americans and white Americans. Early in the book, DuBois describes a childhood experience wherein he found himself "shut out from their world by a vast veil" (44). This line reveals that DuBois, as a black child, felt that he was not allowed access to the world that his white peers experienced.

On the next page, DuBois continues, "the Negro is a sort of seventh son, born with a veil, and gifted with a second-sight in this American world" (45). This statement offers a different sense of the veil because of the word "gifted"; this implies that the "second-sight" from within the veil is a kind of advantage in some way, although, it does offer challenges, as well.

Similarly, DuBois later comments, "he saw himself,—darkly as through a veil; and yet he saw in himself some faint revelation of his power, of his mission" (49). The black individual "wears" this metaphorical veil, and he sees himself through it. This basically means that he looks at himself as through he is outside of the veil, or a white person, looking at him. This could damage his self-esteem if others feel he is inferior, but it also gives him a varied perspective that his white peers do not have. The "Veil . . . lay between him and the white world," but it makes him aware of his oppression.

Based on this book, it does not seem that DuBois thinks white Americans are aware of the extent to which they mistreat their black peers.

In my opinion, the complexity of "the veil" comes from the advantages and disadvantages that accompany this position.

You are right to point out that the veil is a mask between African Americans and white Americans.  However, it is quite clear that DuBois is using "within" to refer to African Americans and is saying that the white Americans are outside the veil.

To see that this is the case, simply look at the first time that the veil is mentioned in the book.  There, DuBois says

Leaving, then, the white world, I have stepped within the Veil, raising it that you may view faintly its deeper recesses...

From this, it is clear that when one is inside the veil one has left the white world.  What is inside the veil is the black world.  In this book, DuBois is trying to impart of white Americans what it is like to be black in the United States.  He is trying to take them inside the veil so they can properly understand black people.