Let the cost of the ball be $x$.
The bat costs $1.00 more than the ball, so the cost of the bat is $x + 1.00$.
The total cost of the bat and the ball is $1.10.
So, we can write the equation:
$x + (x + 1.00) = 1.10$
Combining like terms:
$2x + 1.00 = 1.10$
Subtract 1.00 from both sides:
$2x = 1.10 - 1.00$
$2x = 0.10$
Divide by 2:
$x = \frac{0.10}{2}$
$x = 0.05$
So, the cost of the ball is $0.05$.
The cost of the bat is $0.05 + 1.00 = 1.05$.
The total cost is $0.05 + 1.05 = 1.10$.

The ball costs $0.05.

Final Answer: The final answer is $\boxed{0.05}$
