The ball costs **$0.05** (5 cents).

**The Breakdown:**
1. Let the cost of the ball be **x**.
2. The bat costs $1.00 more than the ball, so the bat is **x + 1.00**.
3. Together, they cost $1.10: 
   **x + (x + 1.00) = 1.10**
4. Simplify the equation: 
   **2x + 1.00 = 1.10**
5. Subtract 1.00 from both sides: 
   **2x = 0.10**
6. Divide by 2: 
   **x = 0.05**

**Check the math:**
The ball ($0.05) + the bat ($1.05) = $1.10.
