A popular version of the Moral Argument goes like this: 
1. If God does not exist, objective moral values and duties do not exist.
2. Objective moral values and duties do exist.
3. Therefore, God exists.
Premise 1 has been traditionally affirmed by many atheists as demonstrated here and it seems reasonable to conclude that in the absence of God, moral values are just the by-product of Darwinian evolution and social conditioning. And if this is the case, as atheist Richard Dawkins says, "The universe we observe has precisely the properties we should expect if there is, at bottom, no design, no purpose, no evil and no good, nothing but blind, pitiless indifference."  Thus, as Chad Williams explains, "...if someone wants to negate the affirmation of premise (1) the burden of proof will lay squarely on them. It will be their responsibility to erect a basis for objective moral values in the absence of God." 
Premise 2 can be demonstrated to the sincere seeker of truth by pointing to some very clear moral truths such as:
1. Torturing people for fun is wrong.
2. Raping someone is wrong.
3. Killing innocent people is wrong.
4. Abusing a child is wrong.
Most will admit that the above are not just socially unacceptable or "taboo," but really, really wrong. We know this from our moral experience.  Again, as Craig states,"People who fail to see this are just handicapped, the moral equivalent of someone who is physically blind, and there's no reason to let their impairment call into question what we see clearly."  For those interested in learning how to handle those "hardliners" that even push back against these very clear moral truths, please see my talk on the subject here.
Please notice that I am using the word "objective" rather than"absolute." This is strategic, as Dr. William Lane Craig explains here:
1. The speakers at Stand to Reason are some of the most clear and concise I have heard.
2. For those curious about what makes a good argument, see here.
3. Richard Dawkins, River Out of Eden, p. 55.
4. Chad Williams, "What is the Moral Argument," http://sealofchrist.com/2011/12/what-is-the-moral-argument/.
5. For more on our moral experience, see here.
6. Ibid., p. 141.