Can I Marry My Sister-In-Law While Being Married To Her Sister?

Imam al-Bukhari and Imam Muslim narrate in their respective Sahihs from Uqba ibn Amir (Allah be pleased with him) that the Messenger of Allah said: 
“Do not go near [non-Mahram] women.”

A person inquired: “What about in-laws?”

The Messenger of Allah (Allah bless him and give him peace) responded, 
“In-laws are death.”

Answer:

In the Name of Allah, Most Merciful and Compassionate. All praise is to Allah, Lord of the Worlds. And the blessings and peace of Allah be upon His Chosen Messenger, the Prophet Muhammad, his family, companions, and followers.

It is completely impermissible to marry one’s wife’s sister. There is clear scholarly consensus regarding this, because of the explicit text of the Qur’an relating to it.

Allah Most High says in the Qur’an, when mentioning those whom it is forbidden for men to marry,

“And (it is forbidden unto you) that ye should have two sisters together, except what hath already happened (of that nature) in the past. Lo! Allah is ever Forgiving, Merciful.” [Qur’an, 4.23]

Imam Abu Bakr al-Jassas al-Razi (Allah have mercy on him) explained in his Ahkam al-Qur’an,

“This entails the prohibition of joining between two sisters in marriage in any way, because of the general purport of the wording.” [Jassas, Ahkam al-Qur’an, 2.186]

Imam Ibn Hazm (Allah have mercy on him) explained in his al-Muhalla,

“This verse means that Allah Most High has forgiven them that which occurred in the past [f: before the prohibition was revealed], not that He has allowed them to stay married [if that was the case in the past]… There is no difference of opinion regarding the prohibition of being married to two sisters at the same time.” [Ibn Hazm, al - Muhalla, 9.134]

Imam al-Kasani confirmed this and explained the reasoning and wisdom behind this prohibition in his Bada’i` al-Sana’i`:

“As for joining between two sisters in marriage, it is forbidden (haram), because of Allah Most High saying, “And (it is forbidden unto you) that ye should have two sisters together…” This is because joining between them would lead to harming family ties [f: between the two sisters], because hostility between co-wives is obvious…” [Kasani, Bada’i` al-Sana’i`, 2.262]

What do you do?

Given the situation, it would be obligatory for you to cut off all relations with your sister-in-law. It would not be permitted for you to talk to her or interact with her in any way whatsoever.  Excessive laxity in interaction, and ignoring the impermissibility of not being alone with people of the opposite sex leads to such problems.

And Allah knows best.

Walaikum assalam,
Faraz Rabbani.