Can you trade Pokemon between a Gameboy Advance and a Nintendo 3DS?

First of all, it's not trade, it's a transfer you'd be doing, I would say one way transfer (always from the oldest gen to the newest).

Even if it's not the question keep in mind that GB to GBA is not possible at all.

Like Chippies said, it's possible via a Nintendo DS (not 3DS), but not all DS games allows you to import Pokémon from GBA, but all games that support transfer from GBA support all Pokémon GBA games.

Here's the full process to make it from GBA to Nintendo 3DS (by using Nintendo DS)

First step : GBA to DS

I will sum up the method describe in this tutorial. (it's the same process with Sapphire, Fire Red, Leaf Green).

This part needs to be done in two process:

1. GBA -> Diamond/Pearl or Soul Silver/Heart Gold

   Once you have finished the DS Game (D/P or SS/HG), you need to go to the Pal Park (in the game) while the GBA game are inserted inside the DS, then you will be allowed to transfer 6 Pokémons every 24 hours (there's a trick on the internet to bypass this limit).

2. Diamond/Pearl or Soul Silver/Heart Gold -> Black & White or Black 2 White 2

   Now that you have all your Pokémon from GBA inside (D/P or HS/GS), you need to transfer it to Pokémon Black/White or Pokémon Black 2/White 2. Once you have finished the DS Game (B/W or B2/W2), you need to go to the Pokémon Tansfer Lab. Fortunately there's no more time limit this time.

Second step : DS to 3DS

This step is much easier, I will sum up the process describe in the official Poké Bank site by Nintendo.

On the 3DS you need to download (via Nintendo eshop) a game called Pokémon Bank. This game has a subscription fee of $5 a year. (If you like Pokémon games, it's worth the price). In your case you can pay the price only for one year. Pokémon Bank is a system that allow you to store your Pokémon on the internet.

When you have Pokémon Bank, you will be able to download another free app (with no subscription) called Poké Transporter. This one will allow you transfer any Pokémon from B/W or B2/W2 to the Pokémon Bank. It's really easy and much simpler than before.

Once your Pokémon are in the Pokémon Bank, you can transfer them to one of the 3DS version of Pokémon : Pokémon X, Pokémon Y, Pokémon Omega Ruby, Pokémon Alpha Sapphire.

The Pokémon Bank will allow you to further transfer Pokémon to Pokémon Sun and Pokémon Moon. Pokémon moved to Pokémon Sun/Moon will not be able to go back to their original game (but you can still put them in the Pokémon Bank).

The Pokémon Bank is also compatible with virtual console versions of Pokémon Red/Blue/Yellow and Green (only available in japan) released in 2016. As for Pokémon Sun/Moon Pokémons can only be transfered to Pokémon Bank but they can't go back to their original game.

For further information check out this article from Nintendo on how Pokémon Bank works with all compatible Pokémon games.

Why such a long process ?

Because Nintendo thinks the process in a generation to generation transfer instead of an 'any generation to any generation' transfer. Moreover, not all generation are compatible between them in term of available Pokémons. Today, things are going much simpler thanks to Pokémon Bank.

What's the shortest path ?

1. Keep your GBA games.
2. Get a Nintendo DS
3. Get Pokémon Diamond (or Pearl) (or HG) (or SS)
4. Finish the Pokémon Diamond (or the one you get at 3.) on the DS
5. Transfer Pokémon from GBA to Diamond
6. Get Pokémon Black (or White) (or W2) (or B2)
7. Finish the game you get at 6.
8. Transfer your Pokémon from Diamond (or the one you get at 3.) to Pokémon Black (or the one you get at 6.)
9. Get a 3DS
10. Get Pokémon X (or Y) (or Omega Ruby) (or Alpha Saphyr)
11. Get Pokémon Bank
12. Get Poké Transporter
13. Transfer Pokémon using Poké Transporter from Pokémon Black (or the one you get at 6.)
14. Use Pokémon Bank to transfer Pokémon to Pokémon X (or the one you get at 10.)

Why would I need a Nintendo DS, Nintendo 3DS can read Nintendo DS games ?

Indeed Nintendo 3DS can read Nintendo DS games, but Nintendo 3DS is not compatible with Gameboy Advance games. Only Nintendo DS is compatible with GBA games. As pointed by king14nyr Nintendo DSi won't do the trick as this console doesn't have a Gameboy Advance Slot

No, it is not possible to trade between 3DS and GameBoy Advanced, mainly because you can't play GBA games on a 3DS natively.
It is possible to transfer Pokemon one way from GBA -> Nintendo DS -> Nintendo 3DS by using supported games as a middle man, but that still won't let you trade between different generations of games.