Well, thanks to the information given by chihiro_ito san in the comment section, I looked up the corresponding webpage:
http://www.goenglish.com/thebestofbothworlds.asp
Based on my understanding, the idiom seems to have a meaning "to get the best of both (excellent) choices".
If this is such the case, then I would say that in Chinese, we have something like「魚與熊掌、不可兼得」, which matches close, but in a negative sense.
So, in terms of Japanese, I searched through Google, and I found the following webpages,
http://zhidao.baidu.com/question/80546257
http://happybibinbatj.blog118.fc2.com/blog-entry-73.html
where they refer to this idiom as "You cannot have your cake and eat it too". With this reference, there comes several translations, such as,
1) 両方よいことはない。
2) 一得一失
3) 一挙両得というわけにはいかない。
4) 両方手に入れるってわけにはいかない。
5) 二兎を追う者は一兎をも得ず。
I cannot tell if the above information may help you well, but I think that you may consider it as a sort of references, since so far no one has provided the best answer.
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Good day. / 良い 一日を。