If he were born in Kenya, the U.S Embassy and State Department would have a record of that birth.
As I said earlier, I was born outside the U.S and have a birth certificate from the Embassy. And even though I was born outside the U.S. I am a natural born citizen.
The State of Hawaii allows residents of Hawaii to register in the state of Hawaii the birth of their children, born in another state or another country, such as Kenya. Please see in attachment a sample Certificate of Live Birth, Box 7C, asks for the County or State or other country of birth.(this provision exists due to the fact that numerous Hawaiians historically worked outside of the state, mostly on Solomon islands and Guadalcanal). This document also has provisions for the name of the hospital and the name and signature of the doctor. Additionally, Hawaii became a State in 1959, only two years before Mr. Obama’s birth. At a time numerous residents of Hawaii did not have birth certificates, therefore the state of Hawaii allowed residents born from 1902 until 1971 to obtain a different document called Certification of Hawaiian birth, that could be obtained simply by the statement made by a relative without any corroborating evidence. Therefore the short version birth certificate, posted by Mr. Obama on his web site, could’ve been based on an affidavit made by his mother or most probably by one of his grandparents, that he was born in Hawaii, even if he actually was born in Kenya. This could’ve been done not because Mr. Obama’s relatives thought that he would run for president one day, but simply to avoid the hustle and expense of going through some eight years of immigration procedures.
Through birth abroad to two United States citizens
In most cases, one is a U.S. citizen if both of the following are true:
Both parents were U.S. citizens at the time of the child's birth
At least one parent lived in the United States prior to the child's birth.
INA 301(c) and INA 301(a)(3) state, "and one of whom has had a residence." The FAM (Foreign Affairs Manual) states "no amount of time specified."
A person's record of birth abroad, if registered with a U.S. consulate or embassy, is proof of his or her citizenship. He or she may also apply for a passport or a Certificate of Citizenship to have his or her citizenship recognized.
Through birth abroad to one United States citizen
For persons born on or after November 14, 1986, a person is a U.S. citizen if all of the following are true:
One of the person's parents was a U.S. citizen when the person in question was born;
The citizen parent lived at least 5 years in the United States before his or her child's birth;
A minimum of 2 of these 5 years in the United States were after the citizen parent's 14th birthday.
A person's record of birth abroad, if registered with a U.S. consulate or embassy, is proof of his or her citizenship. Such a person may also apply for a passport or a Certificate of Citizenship to have a record of his or her citizenship. Such documentation is often useful to prove citizenship in lieu of the availability of an American birth certificate.
Different rules apply for persons born abroad to one U.S. citizen before November 14, 1986. United States law on this subject changed multiple times throughout the twentieth century, and the law as it existed at the time of the individual's birth.
Lots of people fit that category. But BHO cannot. He was either born in the US, or he's not a natural born citizen.