The US law is basically silent on dual nationality...unless the second nationality is obtained after US nationality, in which case the US citizenship can be viewed as abandoned.
One important note is that a dual national is required to enter the US on a US passport. They can use another passport to enter another country, but doing so means they relinquish their rights to embassy assistance in the case of legal problems (as has been in the case in many Chinese government arrests of US citizens).
Note: should your child renounce his US citizenship he will be barred from entry to the US for a 10 year period, last I checked.
Also see here
There are some legal issues about dual citizenship and government services. For example, an American citizen holding another nationality and passport <b>may have difficulty getting a security clearance if that person prefers to use the non-American passport or work within the United States government.</b>
Although being a citizen of more than one country can be helpful as it affords two or more passports, it is prudent to realise that each citizenship carries responsibilites. <b>This may bring about problems in conscription, as well as allegiance to more than one state. A dual citizen is subject to travel restrictions, embargoes and sets of laws issued by multiple governments governing one's behaviour domestically and while travelling abroad. Also, as a drawback peculiar to a few countries such as the U.S., citizens are obligated to pay taxes in both the country of origin and the actual country of residence. </b>However, many countries and territories have contracted treaties or agreements of avoiding double taxation. For example, as at 2005, Hong Kong has already contracted 33 agreements that address double taxation or its avoidance. In extreme cases, such as when the countries of citizenship are at war with each other, a dual citizen's international status can be very complicated.
The number of multiple citizens is large and increasing. Millions of people in the world are now citizens of more than one country. The number of multiple citizens is going to increase rapidly as people become ever more mobile, living, marrying and having children in multiple countries over the course of their lives.
It brings important personal opportunities and responsibilities. As a citizen of a country, you have the opportunity to live there, go to school, work, get medical care, have children, buy property, and retire. There may be agreements to allow freedom of movement to other countries, as in the European Union. (Although it takes just permanent residency to enjoy some of these benefits.) There may also be responsibilities connected with citizenship, such as potential mandatory military service.