This article is from the FAQ, by with numerous contributions by others.
Yes, but note that this is only for renewal or reissuance, not
for obtaining the first visa, and note that this is for the visa
stamp in your passport that relates to travel permission.
Which types of visa can I have reissued in this way? The
recording said E, H, I, L, O, and P. One netter said also A and G.
First step: you have to obtain the form to fill out, by writing to:
Department of State
2401 E Street N.W.
Phone : (202) 663-1225.
You need to say that you want your visa (stamp) reissued, and
would like copies (specify how many) of the appropriate form,
which I believe is Form OF-156. You must also enclose a stamped
self-addressed envelope for them to send you the forms & instructions
Second step: You should receive instructions along with the form.
In summary, the things that need to be sent back to the above
1) Your passport containing the stamp that's to be renewed;
2) The form they sent you (OF-156);
3) The original of your I-94, not a copy;
4) A copy of your petition approval (I-171C or I-797);
5) Letter from employer;
6) Another stamped self-addressed envelope with enough postage
for the return of your passport (courier service could
also be used).
7) A check for the renewal fee [for H1B fee is $100 may have changed]
Six weeks is the normal processing time, or to allow that long.