I have chrome rims on my truck and just had new tires put on. My old tires had no balancing weights on the outside of the rim. When they put on my new tires, they put the weights on the inside and outside of the rim. Can I have these tires balanced with just the weights on the inside of the rim so they don't show on the outside chrome?

 

Yes, But get ready for some compromise. "Single plane" balancing is far less accurate.

 

if chrome rims = alloy wheels, that is why you lost all of your balancing weights. i am yet to come across a shop that will make them stay on alloys.
we had/have 4 cars with alloys and all of them have the same sickness.
so, here's your solution. saving the drowning one is best business of the drowning one, right? so you need to ask the right question.
ask mechanic to put ADHISIVE weights on the rim and COVER THEM with adhisive aluminum foil. here's what you achieve:
1. they never come off
2. they are installed on the inside of rim, onto the INNER FLAT SURFACE of it.
if your mechanic looks at you like as if you just fell from the moon, and ask for pink elephants, let me know, i'll send you pics of this done on my rims. VERY cool. and then take your car some place else, where they know what they doing.
i can personally vouch that they work fantastically well. and i do not have any unsightly weights on the outside of my "snowflakes".

 

Weights stay on my GM Alloy wheels just fine. My GM dealer uses the recommended weights made for these wheels and they are coated with a plastic to prevent any damage to the wheels rim.