I believe you can, but apparently not everyone agrees. Please refer to the feat description:

When using an attack action with a melee weapon, the character can move both before and after the attack, provided that the total distance moved is not greater than the character’s speed.

Note that the feat refers to only a single (attack) action, and does not say the character may/must take an additional move action. The movement described is part of the attack action. In fact, that's pretty much the entire point of the feat. Spring Attack allows you to both move and make an attack with a standard action. Here are some ramifications of having Spring Attack:

1. You can ready a standard action to make a Spring Attack, moving and attacking as that single standard action. This is an enormous boon when melee-fighting spellcasters. That more players don't realize this is a big part of why spellcasters are mistakenly perceived as so much more overpowered than they actually are.

2. In one round, you can move the equivalent distance of a charge and still attack, but you need not do so in a straight line. Also, you don't get the bonuses/penalties of a charge and your effective attack range is 5' shy of your total movement, since you must move that distance after your attack.

If you think that's great, you are correct. Why do you think Spring Attack requires two "tax" feats before you can take it?

So, do you think I'm wrong? Does a Spring Attack require both a Standard Action and a Move Action to perform, or just a Standard Action?

ADDENDUM:

Compare how the Spring Attack feat is written compared to how the Overrun special attack is written, as follows:

A character can attempt an overrun as an attack action made during his or her move action, or as part of a charge.

The Overrun write-up is fairly clear on the point that the special attack is a separate action from the move action. The Spring Attack write-up, however, phrases what most people seem to think is an equivalent mechanism in a very different way. If an Overrun-like interpretation were correct, why not just use equivalent language?

In Pathfinder they describe Spring Attack as a full-round action, and its writeup is much less ambiguous. But Pathfinder is not D&D 3.5.

The thing that gets me is almost everyone who supports the separate "move and attack actions" interpretation agree that the feat as they interpret it is, to put it charitably, "Sub-par". It isn't worth taking two "tax" feats, one of which (Dodge) is almost universally reviled. Yet the "unified move and attack as single action" interpretation makes such worthwhile. Between two possible interpretations of an ambiguous phrase, isn't the one that works better preferable?

It's been put forward that because Spring Attack requires 5' of movement before the attack, with the implication that this 5' of movement can't be part of the Spring Attack action itself. I confess I don't follow the logic of this argument. By my interpretation, both the movement before and after the attack are part of the attack action.

Okay, apparently something printed in the Special Edition of the 3.5 Player's Handbook directly contradicts my argument. I don't have access to it to confirm it, but as much as I hate to lose an argument it's not worth it to me to purchase a copy. So I concede and retract my argument.