Choice A is the correct answer. For exactly one real solution, the discriminant must be zero ($\Delta = 0$).

1. Set $b^2 - 4ac = 0$.
2. $6^2 - 4(1)(k) = 0$.
3. $36 - 4k = 0$.
4. $36 = 4k \Rightarrow k = 9$.

Choice B is incorrect because it is $b^2$. Choice C is incorrect because it is $b/2$. Choice D is incorrect because $k$ must be positive.