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The 'Special Price' section is generated in app/design/frontend/YOUR_PACKAGE/YOUR_THEME/template/catalog/product/price.phtml (this of course can be rolling back to pull from other locations or base/default depending on your theme settings).

In that file there are approximately 7 locations that need edited as each section displays based on different tax settings.

You can find the instances of the special price in that file by searching for

<p class="special-price">

Just be sure to make your edits in all locations in that file that you want to update. Save your changes, and clear your Magento caches.

Updated answer below

This is one of the sections that would need edited in price.phtml (line ~245-273).

    <p class="old-price">
        <span class="price-label"><?php echo $this->__('Regular Price:') ?></span>
        <span class="price" id="old-price-<?php echo $_id ?><?php echo $this->getIdSuffix() ?>">
            <?php echo $_coreHelper->currency($_regularPrice + $_originalWeeeTaxAmount, true, false) ?>
        </span>
    </p>

    <p class="special-price">
        <span class="price-label"><?php echo $this->__('Special Price:') ?></span>
        <span class="price-excluding-tax">
            <span class="label"><?php echo $this->helper('tax')->__('Excl. Tax:') ?></span>
            <span class="price" id="price-excluding-tax-<?php echo $_id ?><?php echo $this->getIdSuffix() ?>">
                <?php echo $_coreHelper->currency($_finalPrice + $_weeeTaxAmount, true, false) ?>
            </span>
        </span>
        <span class="weee">(
            <?php foreach ($_weeeTaxAttributes as $_weeeTaxAttribute): ?>
                <?php echo $_weeeSeparator; ?>
                <?php echo $_weeeTaxAttribute->getName(); ?>: <?php echo $_coreHelper->currency($_weeeTaxAttribute->getAmount(), true, true); ?>
                <?php $_weeeSeparator = ' + '; ?>
            <?php endforeach; ?>
            )</span>
        <span class="price-including-tax">
            <span class="label"><?php echo $this->helper('tax')->__('Incl. Tax:') ?></span>
            <span class="price" id="price-including-tax-<?php echo $_id ?><?php echo $this->getIdSuffix() ?>">
                <?php echo $_coreHelper->currency($_finalPriceInclTax + $_weeeTaxAmountInclTaxes, true, false) ?>
            </span>
        </span>
    </p>

Here is an example of how you could edit it to 1. not display the Regular Price section 2. not display Special Price label

    <?php /*
    <p class="old-price">
        <span class="price-label"><?php echo $this->__('Regular Price:') ?></span>
        <span class="price" id="old-price-<?php echo $_id ?><?php echo $this->getIdSuffix() ?>">
            <?php echo $_coreHelper->currency($_regularPrice + $_originalWeeeTaxAmount, true, false) ?>
        </span>
    </p>
    */ ?>

    <p class="special-price">
        <?php /* <span class="price-label"><?php echo $this->__('Special Price:') ?></span> */ ?>
        <span class="price-excluding-tax">
            <span class="label"><?php echo $this->helper('tax')->__('Excl. Tax:') ?></span>
            <span class="price" id="price-excluding-tax-<?php echo $_id ?><?php echo $this->getIdSuffix() ?>">
                <?php echo $_coreHelper->currency($_finalPrice + $_weeeTaxAmount, true, false) ?>
            </span>
        </span>
        <span class="weee">(
            <?php foreach ($_weeeTaxAttributes as $_weeeTaxAttribute): ?>
                <?php echo $_weeeSeparator; ?>
                <?php echo $_weeeTaxAttribute->getName(); ?>: <?php echo $_coreHelper->currency($_weeeTaxAttribute->getAmount(), true, true); ?>
                <?php $_weeeSeparator = ' + '; ?>
            <?php endforeach; ?>
            )</span>
        <span class="price-including-tax">
            <span class="label"><?php echo $this->helper('tax')->__('Incl. Tax:') ?></span>
            <span class="price" id="price-including-tax-<?php echo $_id ?><?php echo $this->getIdSuffix() ?>">
                <?php echo $_coreHelper->currency($_finalPriceInclTax + $_weeeTaxAmountInclTaxes, true, false) ?>
            </span>
        </span>
    </p>

You will need to make those type of changes commenting out sections or changing their verbiage anywhere in price.phtml that you see 'Special Price' or 'Regular Price'. After saving you may need to clear your Magento caches to see the updates.

If editing the .phtml file gets a bit daunting (and you are okay with just hiding the information from the site user) you could also just do some CSS like:

.old-price { display:none; } /hides Regular Price when there is a Special Price/ .special-price .price-label { display:none; } /hides the words 'Special Price'/