How to translate "虽然~但是" in English?

The English equivalent of 虽然~但是 is "although".  It is grammatically an error to use "although" and be followed by a "but".  In Chinese this is OK to have a "but" following behind.

Also, 但是 is used to contrast between two highly different situations.  不过 should not be translated as "but".  It is the equivalent to "however".

虽然 is equivalent to 'although' and 但是 is equivalent to 'but'. I tried to translate(or select) some sentences. But, in some cases I cannot use the two words, 'although' and 'but', at once. For example:

他虽然不太聪明，但是非常用功 。
Although he is not so clever, he studies very hard.

虽然不想说，但是我赞成。 better to use 不过.  English "but" does not have the shades of nuances as in Chinese.
I hate to say it, but I agree.

虽然下雨，但是/天气/雨水/不冷。 More natural with added words.  In English, "weather" or "the rainwater" is assumed, depending on the larger context.
Even though it's raining, it's not cold.

她虽然很漂亮，但是缺乏自信。  不过 is better
Despite of her beauty, she lacks self-confidence.

(exception)
那个理论虽然新颖，但是还有点儿夹生。
Although the theory is fresh, (but) it is still immature. Would be grammatically wrong in English to have the "but".  So this is not an exception.