Let the cost of the bat be $B$ and the cost of the ball be $C$. We are given that:
$B + C = 1.10$ (1)
$B = C + 1.00$ (2)

Substitute (2) into (1):
$(C + 1.00) + C = 1.10$
$2C + 1.00 = 1.10$
$2C = 1.10 - 1.00$
$2C = 0.10$
$C = \frac{0.10}{2}$
$C = 0.05$

The ball costs $0.05.
The cost of the bat is $B = C + 1.00 = 0.05 + 1.00 = 1.05$
Check: $B + C = 1.05 + 0.05 = 1.10$, which is correct.

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