I agree with both 'Battle Cry of Freedom' and 'A Stillness at Appomattox'
BCoF is the best one volume history I have read.
ASaA is volume three of a trilogy about the Army of the Potomac. The account of Spotsylvania was pretty intense.
A note about the three works mentioned by Little Hawk. Those are works of historical fiction. If that is your cup of tea then have at them. I have heard good things about them, they are said to be pretty true to the real story, but I prefer my history straight. I tried to read 'The Killer Angels' once but didn't like wondering which bits had been invented to move the story along.
Bruce Catton also wrote a trilogy covering the entire war 'The Coming Fury' 'Terrible Swift Sword' and 'Never Call Retreat'
Shelby Foote wrote a narative history trilogy as well that is very readable. Foote's sentiments are with the South, Catton's are with the Union.