Let's first determine how many adult tickets were sold. Given that 2/3 of the tickets sold were adult tickets:
\[ \text{Adult tickets} = \frac{2}{3} \times 345 = 230 \]
Next, we find out how many children's tickets were sold:
\[ \text{Children's tickets} = 345 - 230 = 115 \]
The cost of an adult ticket is $13.95. To find the cost of a children's ticket, which is 2/3 of the adult ticket price:
\[ \text{Children's ticket price} = \frac{2}{3} \times 13.95 = \frac{27.90}{3} = 9.30 \]
Now, we calculate the total money made from children's ticket sales:
\[ \text{Total from children's tickets} = 115 \times 9.30 = 1,069.50 \]
Thus, the theater made \($1,069.50\) from the sale of children's tickets.
The final answer is:
\[ \boxed{1,069.50} \]
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