Yes, Your Grace

This ritual will occur on Week 49 as long as you agree to go forward with it when Aurelea asks you to. If you tell her you won't do it you void the achievement. Once you agree to do the ritual you will be told what ingredients you need and the spellbook. You will need all these items and to perform the ritual incantations correctly.

1 x Glowing Mushrooms - Found in Dungeon
1 x Spellbook - Found in the Council Chambers on the 2nd bookshelf on the far side of the room
2 x Betony Oil - Purchased for 5G/each from the 1st traveling merchant
1 x Mugwort Oil - Purchased for 5G from the 1st traveling merchant
2 x Basilisk Eggs - Purchased for 5G/each from a 2nd traveling merchant

You will begin the dialogue and told to read the book on the chair next to the fireplace to learn the incantation. Read it and tell the Witch you are ready to begin. You will have a timed dialogue choices that you will need to get correct to get the achievement. Below are the correct responses for each question.

1st Choice - We beg for forgiveness for calling you.
2nd Choice - We ask you to give us what we desire.
3rd Choice - Let the mountain break.

Then you will need to put the ingredients into the correct bowls. The witch will tell you which ingredients go in which bowls, and you can have her repeat the instructions or watch the video if you want to see where they go.

2 X Betony Oil in the Old Woman bowl - TOP by Fireplace
1 X Basilisk Egg in the Little Girl Bowl - LEFT
1 X Basilisk Egg + 1 X Mugwort Oil - RIGHT

Then go back to the witch and then some dialogue and then she will say something changed and you need to repeat after her. THESE WILL BE TIMED DIALOGUES.

1st Choice - Nerg va, futaos ferre!
2nd Choice - For ive as tuo me!
3rd Choice - Ring de chiole das err see!

Achievement is yours after the dialogue and you will have a male heir and your wife will be alive. If you mess up just reload. This will also help ensure that you keep your wife alive for the Family Matters achievement.

At week 39 you will be asked if you agree with the ritual. Say: Let’s do it.
You will have to get some items for the ritual. Week 47 is the last week you can get the book from the council chamber if you have left Cedani to keep the bear. It is in the second bookshelf from right to left.
In week 49 you can get the rest: 2 Betony Oil and one Mugwort Oil from the first vendor that comes , 2 basilisk egg from the second vendor and the mushroom from the dungeons. Talk to Aurela in the Bed chamber until you have only this 2 options and then go to the throne room and will be able to end the day – the green and purple icon gets colour – and start the ritual.
At the ritual go to Aurelea and say I’m ready . Then say like this :
We beg for forgivness for calling upon you ;
We ask you to give us what we desire ;
Let the mountain break ;
Place:
2 Betony oil in the middle
Mugwort oil and a basilisk egg in the right
One basilisk egg in the left.
Continue the ritual and say exactly as the witch says for me it was lines 3-1-2. If you got it right the achievement will pop.

A visual representation with commentary of the achievement in the video.