Blacks have always been allowed to become members of the Church. The LDS Church is the only Church which has never segregated its services, and black have been allowed to join in dances and parties. Papers of the day decried the practice in the Church of blacks dancing with whites or even attending those dances. Blacks were even given the priesthood in the early days of the Church, check out Elijah Abel and his sons and grandsons.
I do not know the second president, Brigham Young, I was born years after his death. But I do know that even he stated that one day the blacks would have the priesthood bestowed upon them once more. And his talks about white and blacks marrying were not that. White and blacks could not get married in the south. Those talks were about rape, the white slave master have sex with his black slave women (and girls). Any sexual relations outside a marriage, were considered to be a rape in the eyes of God's laws. And, many of those sexual relations were not consensual. Now I don't know about you, but to me non-consensual sex is rape.
Now I also don't know much about the human climate in the 1800s, since I live in 2012. But I do believe that Brigham Young did know, and based several of his comments on that knowledge. He knew that slavery was rampant in the south and that those same people, who ran the Mormons out of Illinois, would be against any of their slaves joining the Church and being given the priesthood. How do you think they would have reacted had a black priest tried to baptize a white person?
There was a very big misconception of the priestly powers then, and even now. Most believe that power is the same as the priesthood in the Catholic church. It is not, but the ex/anti-Mormons push that belief and many believe it. The fact that any boy, above the age of 12, can hold a priesthood power, and our male missionaries can be called 'elders' even though they are only about 18-20 years old is proof of that.
Edit: @Mike 64. Read that entire chapter. You will find that the quote you give is for the Lamanites, descendants of one Hebrew man, not any blacks from Africa. The Book of Mormon does not refer to any African blacks in its entirety. And the curse was the separation from God, the skin color was nothing more than a marker so that those who were still with God would not marry anyone who was not.
The Book of Mormon even shows that those, whose skin had been darkened, became more righteous than their cousins, the Nephites. So how can the darkening of the skin be the curse?