Yes, a marriage ceremony performed by an official state representative is a valid marriage even though it is not performed in the church. The state does not give the church the authority to marry. It is God who gives the state the authority to marry.
"Let every person be in subjection to the governing authorities. For there is no authority except God, and those which exist are established by God. 2 Therefore he who resists authority has opposed the ordinance of God; and they who have opposed will receive condemnation upon themselves," (Rom. 13:1-2).
It should be clear from the text that God has given authority to the government. Therefore, a marriage ceremony legally performed by government authority is binding. It would not be necessary to go through another ceremony in the church in order to validate the first.
However, if someone wanted to go through a marriage ceremony a second time in the church, simply because they "want to," there is nothing wrong with that. But, we have to be careful that we don't say that the first marriage isn't valid and the second one is because that would violate Scripture.
Also, doubting the validity of a state performed marriage could also mean that a couple, who has gone through such a marriage ceremony but does not consider it valid because it was not done in the church, would then be guilty of sin--not because the state performed marriage is not valid but because they would be considering themselves not married and yet living together.
So, in conclusion, a marriage ceremony performed by the state is valid because God has given authority to the state to perform marriages.
Update (7/17/11): This is not covered in the video, but a marriage performed by the state that contradicts scripture is not valid before God. Therefore, if the state sanctions and performs marriages between people of the same gender, then that is not a valid marriage.