A nice DI box will typically have a thru output.
Route the DI's XLR for direct recording... and route the DI's 1/4" thru output to the Amp.

Most DI's will run on Phantom power too. 
And if the first was passive then there was your problem. Passive DI's will load your guitars PU down so they suck away a bit of juice/ tone.  This is not a problem with active instruments like a synth or acoustic guitar that has a pre amp. But Electric guitars need the active DI. 

That mini cube seems to be a vintage item? cool. 

Your hi impedance 1/4" signal passes through to the amp un altered and the transformer isolates the signal and sends it out Via the XLR as low impedance. This part will be coloured a bit depending on the transformer. 

Most modern amps have built in DI's which are just after the input so will not always be distorted, depends on the amp. 

"Passive DI's will load your guitars PU down so they suck away a bit of juice/ tone."

Hmmmm. I wonder if this applies to bass guitar as well. I've always used passive DI boxes for live and studio work for passive basses.(until I got an interface with Hi-z input.) I'm getting ready to record 6 string leads in exactly the same way as the OP so this post is good food for thought.

I use this http://forum.grailtone.com/viewtopic.php?f=3&t=33585&start=0
 
Uses Jensen transformers.....absolutely pure signal split as many as 4 ways if you wish.  I sometimes  use it to layer my Rocktron Prophesy with TH2, Valvulator and other processors. A bit pricey but one of the best pieces of equipment in my studio.

There is a very simple approach. 

Use a good quailty DI and active at that. Make sure the output that feeds the amp is in direct parallel to the input signal and there won't be any degradation. The high input impedance of the DI box will have no effect on the guitar sound back to the amp.

Use phantom power as well where ever you can with a DI. It is worth it.

Mikes approach while good is a bit over complicated. (It also implies you want the sound of a preamp in line to the guitar amp and in many cases it is not required or preferred. Sometimes with guitar the shortest most direct route is the best option. But Mikes approach also shows a bass guitar not a guitar and for bass I could imagine that would be a nice setup)

In most situations you only need a DI and the mic signal in front of the amp.

If a guitarist has got an array of effects pedals before the amp, a good approach is to record three tracks. One straight out of the guitar as mentioned. (First DI) A second track straight out of the effects units but before the amp (using a second DI) and the third the miced amp sound. Sometimes the effects straight out will sound better than the amped sound with the effects.

I have just done a bunch of guitar tracks and I have found if the guitar is plugged direct to the amp and the amp is very clean sounding you don't even need the DI signal from the guitar either. The clean recorded amp sound also works fine with VST's and things.

But the moment the amp sound is distorted (or effected) in any way then the DI signal becomes rather important and useful.

Spend the time getting the amp settings correct and you will be much happier. Check the guitarist is using the best pickup and settings for any given sound. (and part) Often I find they don't always extract the best possible sound and you might have to do it for them. A professional guitarist will let you fiddle the sound if you are not happy. 

Keep the amount of reverb in the amp way down. Guitarists often try to wind up the reverb. It is horrible usually and you are stuck with it forever. You can add in any spring reverb later but also much nicer reverbs too.

If you are doing multiple parts and only have one amp, having several guitars is a fantastic option. Get them to switch guitars. Even better is if they have several amps and several guitars. The approach of changing guitars and amps will have a far greater impact later on in mixing where you might want to separate things a little.

Another thought

Use the effects send, if your amp has one, and go direct with that. You won't need the pre-amp portion of the amp sim. Just the cab portion.

Be wary of using the effects send of any guitar amp. Not good advice. Some amps that have a distortion channels like Marshalls might be sending the distorted signal from there.

The ultimate place to get a total clean sound is direct out of the guitar into a DI. Then you know you have a perfect clean and unaltered sound.

.

Thanks, lot's of good info in here! My mini cube worked fine.

One important discovery was as opposed to using a line out or fx loop, is that with the DI, or split being placed right after the guitar, I didn't have to work about the pedal that were placed in use.

I never thought about using phantom power on a DI, it really works? I assumed since it takes a 9 volt battery that the 48 volts would not be healthy...   Any input on that, how or why it would work from a theoretical view?  

.