ISO boy/girl born in Houston, 1967/68

I have just recently discovered (through my older sister) that my mother gave up a child that she had sometime in 1967 or 1968. This was kept secret in our family. I very much want to find this missing sibling and learn how they are, learn if they would like to be contacted by their brother or have other questions. My mother and I are not on speaking terms so I am unable to find more specific information. I do know that this person was born either the later part of 1967 or early 1968. My sister believes that this person is a girl but is not certain. The child is white. My mother at the time would have been in her late thirties (37 to 39). Her maiden name is Brickey and her last name (married name) was Alexander. The child is likely to have been born in the Houston area but might also have been born in Dallas (her father lived there at the time). Frankly, I'm still somewhat in shock that this could have been kept a secret from me for so long. I really would like to find this person and let them know that they have a family if they wish to find it. Any suggestions for how to go about my search are much appreciated! Doug