I wasn't keen on using the Linq solution above so I wrote this:
/// <summary>
/// Takes a datatable and a column index, and returns a datatable without duplicates
/// </summary>
/// <param name="dt">The datatable containing duplicate records</param>
/// <param name="ComparisonFieldIndex">The column index containing duplicates</param>
/// <returns>A datatable object without duplicated records</returns>
public DataTable duplicateRemoval(DataTable dt, int ComparisonFieldIndex)
{
    try
    {
        //Build the new datatable that will be returned
        DataTable dtReturn = new DataTable();
        for (int i = 0; i < dt.Columns.Count; i++)
        {
            dtReturn.Columns.Add(dt.Columns[i].ColumnName, System.Type.GetType("System.String"));
        }
        //Loop through each record in the datatable we have been passed
        foreach (DataRow dr in dt.Rows)
        {
            bool Found = false;
            //Loop through each record already present in the datatable being returned
            foreach (DataRow dr2 in dtReturn.Rows)
            {
                bool Identical = true;
                //Compare the column specified to see if it matches an existing record
                if (!(dr2[ComparisonFieldIndex].ToString() == dr[ComparisonFieldIndex].ToString()))
                {
                    Identical = false;
                }
                //If the record found identically matches one we already have, don't add it again
                if (Identical)
                {
                    Found = true;
                    break;
                }
            }
            //If we didn't find a matching record, we'll add this one
            if (!Found)
            {
                DataRow drAdd = dtReturn.NewRow();
                for (int i = 0; i < dtReturn.Columns.Count; i++)
                {
                    drAdd[i] = dr[i];
                }
                dtReturn.Rows.Add(drAdd);
            }
        }
        return dtReturn;
    }
    catch (Exception)
    {
        //Return the original datatable if something failed above
        return dt;
    }
}
Additionally, this works on ALL columns rather than a specific column index:
/// <summary>
/// Takes a datatable and returns a datatable without duplicates
/// </summary>
/// <param name="dt">The datatable containing duplicate records</param>
/// <returns>A datatable object without duplicated records</returns>
public DataTable duplicateRemoval(DataTable dt)
{
    try
    {
        //Build the new datatable that will be returned
        DataTable dtReturn = new DataTable();
        for (int i = 0; i < dt.Columns.Count; i++)
        {
            dtReturn.Columns.Add(dt.Columns[i].ColumnName, System.Type.GetType("System.String"));
        }
        //Loop through each record in the datatable we have been passed
        foreach (DataRow dr in dt.Rows)
        {
            bool Found = false;
            //Loop through each record already present in the datatable being returned
            foreach (DataRow dr2 in dtReturn.Rows)
            {
                bool Identical = true;
                //Compare all columns to see if they match the existing record
                for (int i = 0; i < dt.Columns.Count; i++)
                {
                    if (!(dr2[i].ToString() == dr[i].ToString()))
                    {
                        Identical = false;
                    }
                }
                //If the record found identically matches one we already have, don't add it again
                if (Identical)
                {
                    Found = true;
                    break;
                }
            }
            //If we didn't find a matching record, we'll add this one
            if (!Found)
            {
                DataRow drAdd = dtReturn.NewRow();
                for (int i = 0; i < dtReturn.Columns.Count; i++)
                {
                    drAdd[i] = dr[i];
                }
                dtReturn.Rows.Add(drAdd);
            }
        }
        return dtReturn;
    }
    catch (Exception)
    {
        //Return the original datatable if something failed above
        return dt;
    }
}