To duplicate the salinity of seawater using regular table salt, how
much salt would I need to dissolve in one gallon of water?

Since the salinity of seawater is typically 3% to 3.5%, to approximate
the salinity of seawater, your ratio should be 30 to 35 parts salt to
1000 parts water.

Eight tablespoons of salt (4 fluid ounces) added to one gallon of
water (128 fluid ounces) will give you a good approximation of
seawater.

Keep in mind that seawater contains trace elements, additional
minerals that are not present in table salt (sodium chloride), so this
won't actually duplicate seawater closely enough to be suitable for a
saltwater aquarium.

"Seawater is about 3% salt - 1/2 ounce of salt in one pint of tap water."

Florida Aquaculture: Teacher Tips
http://www.floridaaquaculture.com/aquaed/Teacher%20Tips.htm

"Seawater contains 3 percent salt by weight; this is equivalent to 30
parts per thousand (ppt) or 30,000 parts per million (ppm)."

The Use of Salt in Aquaculture
http://article.dphnet.com/cat-02/salt.shtml

"The average salinity of sea water is 35 part per thousand by weight
(3.5 percent by weight)."

Advanced Water Systems
http://www.advancedh2o.com/technical/body_sea_water.html 

"There are many different components of sea water. However, there are
eleven ions that make up its main chemicals. Those eleven chemicals,
according to Oceans by Karl Turekian, are (in grams per kilogram of
water): Chloride (19.l35), Sodium (10.76), Sulfate (2.712), Magnesium
(1.294), Calcium (0.413), Potassium (0.387), Bicarbonate (0.142),
Bromide (0.067), Strontium (0.008), Boron (0.004), and Fluoride
(0.001)...

Chloride and sodium are the elements in table salt. That is why
seawater is referred to as salt water. The two most abundant chemicals
in the sea make up salt... If you dissolved 35 grams of salt that you
would find in your house in one liter of tap water, you will get
something that is practically seawater."

Components and Salinity of Seawater
http://drake.marin.k12.ca.us/stuwork/rockwater/the%20salt%20in%20seawater/saltinseawaterpg3.html

Here you'll find a detailed look at the mineral components of seawater:

Sea Friends: Composition of Seawater
http://www.seafriends.org.nz/oceano/seawater.htm

Google search strategy:

Google Web Search: salinity seawater percent
://www.google.com/search?hl=en&ie=UTF-8&q=salinity+seawater+percent

I hope this helps. If anything is unclear, or if a link doesn't work
for you, please request clarification; I'll be glad to offer further
assistance before you rate my answer.

Best regards,
pinkfreud

Thank you, thank you. Perfect answer in a record time. Much appreciated.

Pink,

You are truly amazing.  I offically declare myself your #1 fan

I am also Pink's # 1 fan !!!!!!!!!!!!

sponky,

Thank you very much for the five-star rating and the nice tip!

~pinkfreud

If you dissolve 8 tablespoons of table salt in 1 gallon of water, the
salinity will be about 39 parts per thousand, which is saltier than
ocean water.  The reason is that salt is heavier than water, so if you
have a solution that contains 3% salt by volume, it contains more than
3% by weight.  The density of table salt varies from 1.2 to 1.3 times
the density of water, depending on crystal size, structure, etc.

http://www.saltinstitute.org/teaspoon.html

A better approximation is 3 1/4 tablespoons salt per gallon.

Oops, I mean 3 1/4 fluid ounces.  That's 6.5 tablespoons per gallon.