Did God allow a believing man to marry only one believing woman with the exception of orphan girls provided that they will be treated equally? 

 

God recommends a believing man not to marry more than one orphan girl (or non-orphan girl) considering the difficulty in treating them equally!

 

Verses 2-3 of Chapter 4:

And give the orphans their wealth and do not exchange the pure for the foul; and do not devour (or use up) their wealth, mixing it with your own; this is indeed a great sin. And if you fear that you will not be just towards orphan girls, marry the women whom you like - two at a time, or three or four; then if you fear that you cannot keep two women equally then marry only one or the bondwomen you own; this is closer to your not doing injustice.
 

Verse 20 of Chapter 4:

And if you wish to change one wife for another and you have given her heaps of treasure, do not take back anything from it; will you take it back by slander and open sin? 

Verse 129 of Chapter 4:

Ye are never able to be fair and just as between women, even if it is your ardent desire: But turn not away (from a woman) altogether, so as to leave her (as it were) hanging (in the air). If ye come to a friendly understanding, and practise self-restraint, Allah is Oft-forgiving, Most Merciful.