Amber Rose Having Rough Early Pregnancy, Same Sickness as Kate Middleton

Amber Rose is having a rough early pregnancy ... because she has the same sickness Kate Middleton had to endure.

Sources close to Amber tell TMZ ... she's 13 weeks pregnant and is suffering from hyperemesis gravidarum ... a severe type of nausea and vomiting during pregnancy. We're told Amber was hospitalized at Cedars-Sinai Medical Center for a few days, getting IV treatments because she was sick and losing weight.

It's not all bad though ... we're told Amber and her boyfriend, Alexander "AE" Edwards, are thrilled about having their first child together. We're told the pregnancy wasn't planned, but they weren't taking precautions because they're deeply in love.

The baby boy is due in October, close to Amber's birthday ... and we're told she's excited to share the same sign -- Libra.

Amber publicly announced her pregnancy Thursday, and we're told she had a gender reveal party at her house a couple of weeks ago ... but it was purely a family affair.

Kate battled the condition through all 3 of her pregnancies ... and now Amber's had it for both of hers.