Hello Customer (name blocked for privacy), thank you for your question. You would be eligable for a temporary work visa right away. The first thing you must have for a temporary work visa is a letter of acceptance. Your employer must furnish you with a piece of correspondence - preferably on corporate letterhead - formally offering you a position with the company. In this letter, your employer should note the title of the position and its responsibilities, the salary, the start and end dates, and how many hours a week the position involves.
Your employer will then need to fill out an application on an INS Form I-129, petitioning for permission to hire you as a temporary employee. Of course, in all practicality, this means that you fill out the form for your employer and then hand it to them for their signature. The form can be downloaded here: http://www.ins.usdoj.gov/graphics/formsfee/forms/i-129.htm
Then you wait. The INS can take several months to process these requests. Once you receive your visa, the government will instruct you on the mechanics of activating it, which (if you are already in the United States) may involve leaving the country and visiting a U.S. Embassy abroad. When you travel on an H visa, you will not have an I-20 or an IAP-66, but the INS will furnish you with proof of your visa, and it is always a good idea to carry your employer's letter of acceptance with you.
Additional information can be found at the INS website: (http://www.ins.usdoj.gov/)
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