Let's denote the cost of the ball as x. 

Since the bat costs 1.00 more than the ball, the cost of the bat is x + 1. 

Given that the total cost is 1.10, we can set up the equation: 
x + (x + 1) = 1.10
2x + 1 = 1.10
2x = 0.10
x = 0.05

Therefore, the ball costs 0.05.
