death and new baby

So I've just found out I am pregnant with my second baby. The kicker is I found out on the same day that my little sister died so the only person I've told besides my husband is my brother. I worried to tell my mum because I think it's too much for her to bear with just losing her youngest daughter and I don't want her to feel obligated to be happy about it in such sensitive circumstances. Also the baby is due the same time of my sisters birthday. I'm extremely worried about having a baby so close to my deceased sisters birthday as my mother is an emotionally unstable person and I think it will tip her over the edge. I feel completely lost on how to handle this delicate situation.
I love my sister and miss her terribly but I'm not getting rid of my baby because of her death. How should I approach my mother about this?