Yes, this is what we know as gestational surrogacy, host surrogacy, or IVF surrogacy, and the woman who carries another woman's egg is a gestational carrier (GC) or gestational surrogate. Note that the name IVF surrogacy means that in vitro fertilization (IVF) is the technique of choice when the surrogate is not genetically linked to the child.
For this to be possible, it is necessary that the intended mother or an egg donor, if the intended mother cannot use her own eggs, undergoes follicle puncture to have her oocytes retrieved. Then, they are fertilized in the lab with the husband's, partner's, or donated sperm to create embryos.
Finally, the resulting embryo or embryos are transferred to the uterus of the gestational carrier. So, actually, what the GC carries is an embryo or embryos that have been created using the gametes or another person or persons, but not an egg.
