The first sentence is correct and the second one is not correct.
That is because, 可愛い is so-called 形容詞 (i-adjective).
The second type of conjugation is correct for 形容動詞(na-adjective), but not for 形容詞.
○:この犬は可愛くないです。
×:この犬は可愛いではありません。
Instead,きれいだ is 形容動詞(na-adjective), so the second type of conjugation fits in.
×:彼女はきれいくないです。
○:彼女はきれいではありません。
You may notice, you can replace na-adjectives with nouns and still the same conjugation holds.
×:彼女は学生くないです。
○:彼女は学生ではありません。
Roughly speaking, older words are classified into 形容詞 (the first expression) and therefore it conjugates more naturally as adjectives, and newer words 形容動詞(the second expression) are sort of more clumsy, so it nearly conjugates as just nouns.
(although there are exceptions as in most grammatical rules)
related posts:
大きじゃない vs 大きくない?
Why does Japanese have two kinds of adjectives? (-i adjectives and -na adjectives)^