Which of the following sentences is correct, or are they both correct?
I'm not so cold-hearted that I wouldn't reply.
I'm not so cold-hearted as to not reply.
Both are completely acceptable and mean essentially the same thing. You can choose based on personal preference or context.
There appears to be a nuance, making the two sentences quite different in their focus.
The use of that I returns the focus to I. The speaker is the focus of the sentence. 'Don't think I am that bad.'
The use of the form so[phrase]as concentrates on the [phrase]. Here, cold-hearted is the focus of the sentence. The speaker is not drawing attention to himself but discussing about someone being cold-hearted.
Such constructions are sometimes useful in writing to show what can be expressed in speech by a mere change in the stress.