Let's denote the cost of the ball as \( x \) dollars. According to the problem, the bat costs \( 1.00 \) more than the ball, so the cost of the bat can be expressed as \( x + 1.00 \) dollars.

The total cost of the bat and ball together is given as \( 1.10 \). We can set up the equation:

\[
x + (x + 1.00) = 1.10
\]

Simplifying this equation gives:

\[
2x + 1.00 = 1.10
\]

Next, we subtract \( 1.00 \) from both sides:

\[
2x = 0.10
\]

Now, we divide both sides by \( 2 \):

\[
x = 0.05
\]

Thus, the cost of the ball is \( 0.05 \) dollars, or \( 5 \) cents. 

To confirm, if the ball costs \( 0.05 \) dollars, then the bat, costing \( 1.00 \) more, would be \( 1.05 \) dollars. Together, that totals:

\[
0.05 + 1.05 = 1.10
\]

The calculations hold true. Therefore, the cost of the ball is \( \boxed{0.05} \).
