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According to the progressive Jewish movements (Reform and Reconstructionist), a child with only one Jewish parent--either mother or father--is Jewish as long as the child is raised to identify as Jewish. Holding a bris or simchat bat for your infant can be the first step to raising the child Jewish.
According to the traditional Jewish movements (Orthodox and Conservative), a child is not Jewish unless he or she has a Jewish mother.
Mohels trained in the Reform tradition will perform a brit milah for children of interfaith parents without the expectation of further steps to conversion.
Many mohels are Orthodox, however, and therefore abide by the traditional definition of a Jewish child. In many cases they will agree to perform a circumcision for a mother who is not Jewish with the understanding that the child will later be immersed in a ritual bath (mikvah) to be converted to Judaism. If the father is not Jewish but the mother is, the mohel will skip the Hebrew line from the traditional brit ceremony where the father delegates his responsibility to circumcise the child to the mohel.
Generally speaking, mohels are quite accommodating to the needs of parents, so if you would like to have a grandmother or relative who is not Jewish to be involved in the ceremony, don't be afraid to include them. In rare cases a very traditional Orthodox mohel may insist that the sandek, the man who holds the baby, be Jewish.
For the simchat bat, there are fewer concerns because there is no standardized procedure or officially recognized officiant. For any kind of birth ceremony, it is valuable to create a program that provides a guide to the rituals for both those who are and are not Jewish. If you don't have time to create a program, it's helpful to explain the ceremony in advance to relatives that are not Jewish.
Also be aware that if you plan on having the brit milah or simchat bat at a synagogue, there may be restrictions on what people who are not Jewish can or cannot do. Some Conservative congregations, for example, may not allow the parent who is not Jewish on the bima. Discuss the congregation's policies with the rabbi of the congregation before holding a brit milah or simchat bat so that there are no surprises. Since you only have a few days to plan these ceremonies, it might be worthwhile to speak to the rabbi before the child is born.
If you plan on affiliating with a Conservative or Orthodox congregation and the mother is not Jewish, the child must be officially converted before the congregation recognizes him or her as a Jew. For boys, the brit milah is part of the conversion process. For both boys and girls, a Conservative or Orthodox rabbi will also typically require the child to be briefly immersed in a ritual bath known as a mikvah. Conservative and Orthodox rabbis will also require an official conversion of adopted children whose mothers aren't Jewish or whose heritage cannot be verified.